Use scouts

Digging and building mineshafts is very easy to detect for even a casual observer. The BBEG can wait until the druid has wasted a bunch of his spells, then show up and bury the intruders in the grave they just dug. If it's an ambush, the PCs won't be able to teleport away without taking a few hits, and if they have a reputation for being flighty, the BBEG is likely to use poisons, diseases, curses, and other ongoing problems that won't be solved by a quick rest.

Use consequences for a slow approach

When the BBEG sees the PCs digging, he can predict where they'll end up and trap/reinforce that area of the lair. The PCs break into the lair, and it's flooded with neurotoxin or acid. This goes double if the PCs run away and then try to come back - the area will be swarming with minions, there will be defenses erected, and so forth. The BBEG may send his own spies out to find the PCs' base (or things they care about) and issue a retributive strike.

Use their reputation against them

Digging big ol' holes in the ground is far from business as usual in the realm of fantasy warfare. Stories of their weird tunneler types will spread, and evil overlords everywhere will defend their bases against this approach. The simplest solution is to bury explosive devices in multiple layers, so that the digging animals are killed and the druid needs to use another spell.

Use special terrain

Digging in dirt is easy. Digging through mud is easy, but maintaining the shaft afterwards is hard. Digging through granite is all but impossible. Digging through lava gets you dead. Digging into another plane or into a sky fortress makes no sense. If there's an underground river, digging underneath it becomes a challenge - you need to keep all that water out.

Use subterranean monsters and lairs

All those badgers are coming from somewhere. Some of them may have already done some digging - and instead of landing into the BBEG's lair, the party lands into a giant badger warren, or a formian hive, or the Underdark.

Use skill checks

Mine shaft structuring is a complicated business; in real life, it's a job people go to college for. Does your druid have structural engineering knowledge? No? Then he put the support in the wrong place and the mine shaft collapsed.

Use third parties

There are more people in the world than the BBEG and the PCs. What if the PCs are digging in the middle of a sacred elven grove, and the elves come and ask them to stop? What if all their digging attracted another adventuring party, who decided to storm the citadel the old fashioned way and has already cleaned up by the time the PCs break in? What if the PCs are digging in crown lands, and a magistrate comes and demands to see their zoning permit?

Talk to the Players

They are used to being the target of a DM out to kill them. You're a DM trying to let them be the hero. There is still a risk of dying, but you aren't trying to kill them like their former DM was. Tell them that they can, and are encouraged, to try new things and take risks. Specificially talk about their spell and say "It was cool, but I'll be actively discouraging you from using it that way again."

Obsticles that Make it the Dangerous Option

Suggestions that come to mind:

• Make the BBEG's base surrounded by water or very high tower.
• Have a large perimeter around it crawling with gaurds.
• Have a lava flow close the the surface that he exploits for smithing his minion's arms (ala LOTR)
• Let the BBEG know about it and have him devise a trap tailor made to that kind of attempt.

Play the Conjure Animals Spell RAW

Conjure Animals, as written, lets the players pick the level/number, but the DM picks what actually shows up. They can try to get something that digs, but you could make birds appear instead. This change would be different from what you've been playing, then you could run into people problems ("Aw, Com'on! You let us before!").

They say there are things in the earth that should remain buried.

Do you remember the fate of the Dwarves of Moria?

Something could be unveiled; something wrong that changes the dynamic of the setting/party and puts them out of their comfort zone. Not as players, but as characters in the game.

In my head I have some weird quest hook - the druid's animals unveiled a potent ancient plague and became infected. What will the druid do and feel, knowing that he is the sole cause of their suffering? What will happen to the world if an old plague takes hold of it now, when modern creatures are susceptible?

The implication would not be a TPK but something really unsettling and world-changing.

Another way to handle it would be to let players have their way. They might get bored and change their tactics to make the game more enjoyable.

How about letting your BBEG react to the players' plans? Perhaps he has the means to counter their attack vector? I don't know your BBEG's capabilities so I can't give more advice here.

But I have a horrible picture in my head: the druid digs a pit with help of his squirrels, but the BBEG sends a 'squirrel-vampire' (an undead creature that can replicate by killing/turning other squirrels, or otherwise sabotage the workforce {mind-controlling fungus, anyone?}) and it cripples the excavation efforts, while also creating a threat that grows exponentially. What could the players do to prevent a disaster that can wipe out their digging force?

You could use fantasy terrain that is actively dangerous for digging. For example, earth mixed with broken glass, a literal Glass Field, or a swampy area where any tunnels flood quickly.

As a side-note, has the druid thought about his impact on the ecosystem? His mass diggings are sure to impact the normal lives of forest creatures. What the hell, hero?!

I have faced similar situations in the past - specifically that I had a group of over-the-top engineers who actually knew a lot about the various topics/subjects that came up. This typically lead to a "this is possible, and within the rules" conversations quite often.

However don't be tricked into following their mindset. You are the DM, by the very nature of the game you should be controlling the flow of events/story. Thats not to say level with the players and be compromising, an ingenuous solution to a tough problem should be part of the experience. Here is some ideas that should help you get through this and still give a great story experience.

1. Just because it is "possible", does not mean it is feasible or safe. I hate to say it - but unless your druid is en expert miner with a background of experience and knowledge his tunnel is probably going to collapse, or at least have a variety of problems.
2. Underground monsters. There is a lot of them, if you feel the party is going to make the entire encounter useless just start throwing monsters at them until they deviate back onto the correct path. Typically I use a factor of 1.5x depending on the distance they are from the main plot to keep them in line (e.g. The first time they step out of line the encounter is 1.5x harder then it was supposed to be, if they continue down that path continue adding to the difficulty and trying to encourage them to backtrack to the proper path).
3. Natural disasters (earthquake, flood, etc). A little heavy handed but entirely possible.
4. Have very hard deadlines known to the players. Things like, "the princess will be killed if you take more than 2 days to rescue her, and the bandits leave with their loot in hand. It is sad for your story, but the players will learn that every quest from that point on should be followed strictly rather then taking their time.
5. Lastly - Just do not allow it. When they pitch the idea simply tell them, "it will not work". They may ask for a reason but you simply have to say something along the lines of "Try it and find out, but it will be a waste of time". This can also be done through in-game RP (such as saying the fortress is magically protected from access other then the front doors).

Talk to your group to develop a mutual understanding and set campaign expectations.

You seem to understand a lot of causes for the issue at hand. Tell your group that you understand they're risk-averse but that you're not interested in killing them. Except in extreme circumstances or their total disregard of explicit risk, a TPK is a failure on your part; let them know that you feel this way. Your bad guys can have motivation for capture over killing.

Let them know that you truly appreciate the first time this plan was used. It was resourceful, creative and interesting. However, let them know that you don't want to design a campaign specifically to be bypassed using the same technique, and that such techniques are limited use.

Ask if they want the campaign to become about developing this technique, and your world developing answers to it. If so, start gradually implementing countermeasures as discussed in other answers.

Regarding the deadlines issue, this will be hard behaviour to change immediately, even if they start taking more risks.

Deadlines can be a really interesting way to force players to prioritise and make tough decisions. You could try running sessions that highlight this; for example having a time counter (a d20) tick up every round and keep actions within rounds outside of combat. If somebody gets their leg stuck, the party has to decide whether to spend a tick helping them out or let them arrive late to the critical moment.

Alternatively, you could incentivise sticking to deadlines by having them affect the players. Oh, now it's not just that the princess was captured, but the party is somehow implicated and will be held responsible if they can't fix it. Oh, now a party member has a curse that's gradually worsening.

Or you could just have the rewards become unreachable over time. After a deadline, the bad guys pack up and move on, or they're gradually consuming/destroying a scaling reward.

What's the Real Problem?

Now, technically, you're talking about digging here, but you mention that your players ignore deadlines.

It's not that hard to make digging hurt; you can set up wherever they dig into to be more or less equivalent to the gatehouse they would storm if they went in the normal way, for instance.

However, what you seem to really be having issues with here are players that want to focus on interesting exploits, rather than traditional narratives.

Now, this could have its own issues associated with it: players could want a different genre, or feel that this is a way to get agency that they aren't getting through more traditional applications of their character, but I'm going to go ahead and assume that those aren't the issues.

Make Time Matter

You mentioned that your players blow off timelines. Don't let them do this. Now, obviously you can't force them to do stuff; teleporting them into a boss fight with the dude what needs killing is only fair or interesting once or twice tops, and you need a fair deal of setup to get that to work. Likewise, having the objective they need to complete come to them is going to be problematic as well.

With the tunneling thing, there are tons of things you could do to remove the tunneling, but this could lead to other exploits. What if the players are content to summon giant eagles and bombard the fortress with boulders?

If they don't meet the deadline, the quest fails. They don't get a reward, because the reason they were sent to stop the guy has already become outdated. Either the BBEG's plans succeeded, the quest-giver died at the hands of an assassin, or they simply get scolded for taking too long.

Communicate Expectations

One thing you mentioned is that your players are used to a very brutal DM who just killed them a lot. You need to make clear that you are not that way. You can suggest things like Schrodinger's Dice, where players roll for their characters' deaths in secret and are never challenged on the dice: if they go down in a fight they'll be alive unless something finishes them off (if they desperately want it enough to cheat), and you control the bad guys so they won't be finished off.

Someone in comments mentioned the Same Page Tool; there are a lot of other ways to do this, but basically have a heart-to-heart with your players and mention your concerns with the group's planning. It may turn out that they like shenanigans like this, and then you have to decide whether or not to allow it or to encourage them to move away from it.

Emergent vs. Scripted Events

One thing to consider as well is the nature of the game: role-playing games offer an incredible amount of flexibility in actions and storytelling, and emergent play (in which outcomes are based on freely selected actions) often is a result, as opposed to scripted play (in which outcomes are based on a limited selection of actions).

Talk to your players, and ask them if they'd be fine playing more genre-traditional strategies. If not, recognize their emergent play and make it part of the setting. Enemies learn from their mistakes: if the characters let their secrets slip, or let any of the BBEG's minions survive to tell the tale, every villain they encounter has planned for their schemes. Let them know this: each strategy only works once.

Part of this is pushing the deadline. If the villain knows they'll take fourteen days to tunnel into his fortress, the appropriate response might mean setting off the doomsday device on day seven. Be fair with this, of course, and make sure that the players know that there is urgency. Heroes can say no to the mission parameters: they're not typically soldiers (though if they were you could throw insubordination at them) of the crown or anything like that, but they can't say no to harsh realities of their enemies' victories.

Every 10 feet of digging, perform a secret D20 roll. Add +1 to the result for every 10 feet of total depth. Only announce when something happens.

1-12  nothing happens
13-15 change in terrain (change at random to solid rock, underground lake, mud, gravel, lava, etc)
17-19 minor accident (rock fall, flood, minor collapse etc) players must make a save or take some damage
20    animal encounter - players had dug into the burrow of a (now furious) underground animal    
21    booby trap - players trigger a mine or dig through the roof of a spike pit
22    total cave in - players must make some sort of save or be crushed to death

Details of the list don't really matter. As soon as you start rolling they will realise something is going on. As soon as you announce your first result they will figure it out and be very wary of going any further.

I'd be wary of just hitting them with something really nasty straight off. You don't want to be like their previous brutal DM and tpk them just for trying something out. Best to put the frighteners on and persuade them off their current course of action.

Problem: Your players expect you to both try to kill them with numbers and be an unimaginative GM who does not react to their actions in any way (aka a bad GM).

Solution: Have the bad guys make mistakes.

If you give the players information on what is going on - overhead conversations, diagrams of observed patrol routes, enemies living in mountain chalets with big windows and ineffective walls, that information can reveal vulnerabilities in security design (intentional ones on your part) that will be taken as opportunities by the group. They may ignore a few - thinking them 'traps' designed to 'kill them and it's their fault' so you can gloat about it or whatever bad GM 'orb of annihilation door handle' behaviour they've come to expect - but eventually they'll take one, and they'll feel smart and like badasses and get addicted to doing that, and the 'exploit' style behaviour of treating the game like a CRPG scenario they have to 'break' will quickly disappear in a tide of taking the world as real.

Once someone has their 'trust' in the GM shattered, you need to rebuild that trust by tricking them into trusting by accident and then rewarding that trust with a fair and realistic outcome.

Play God

According to some of the flavor text in the PHB for druids (pg 64), druids get their power either from a deity or nature itself. Druids strive for balance in nature. Depending on how in line with the DnD lore your players' character is, you may be able to justify a few loss of powers.

What your player is doing is moving literal tons of earth around. This can be argued to upset the delicate ecosystem in which they are tunneling. This is a big no-no for druids generally. Use this as justification to take away your druid's abilities temporarily.

Other answers have suggested numerous defensive mechanisms to protect against tunneling, but I am suggesting you eliminate it altogether. Though, I would advise you only restrict your players if they keep doing the tunneling.

My experience with this type of thing: it creates some tension in the group and also allows for a lot of roleplaying chances. Your druid attempts to conjure animals and nothing happens. Why? Well now they have another quest on their hand, which creates more opportunities for storytelling. Of course, now they must redeem themselves in the eyes of their deity. More fun times!

One final, very important note:

Talk to your players

You really need to let them know what kind of campaign you're running and what kind of deadlines they're facing. You could even let them know via some divine intervention, which could be combined with the deity explaining why his powers are missing. How can they know the tunneling is too slow unless you tell them?

When the party wants to do something that is going to take a long time, and you don't want to have to sit there while they try it, you can just use a timeskip and jump over it.

Player: "We're going to dig. So, I start by casting-"

DM (interrupts): "Right! You spend the rest of the day digging. It looks like it will probably take another week to get through. You want to keep going?"

Player: "Yes, we-"

DM: "Ok, you dig through the week. Eventually you make it through. You are now standing at the opening where your tunnel joins the underground lair"

And then keep playing from there, just as if the players had walked in the front door. Or, if it is missing the deadline itself that is the problem, you can use one of the other suggestions here - like maybe you say

"At the end of the week you break through into the underground fortress, just in time to see a party of adventurers walk by, celebrating their victory over [badguy]. Gee, if only you had gotten there sooner..."

That way you can jump over the players messing around, and get straight to the next adventure. If the players keep doing this, by the end of the play session, they may have tunnelled into a dozen different lairs, and been too late every single time. I think by that point, they might just take the hint!

Remember that as the DM you can simply tell your players that they can't do something. A variety of rationales exist for why they couldn't build a tunnel into a dungeon.

• The ground is unsuitable for tunneling (Rock, Sand, Mud, etc.).
• There are wards on the dungeon that prevent tunneling in. These wards could predate the current inhabitants and thus not require casting by the current inhabitants.
• Ask them to make a wisdom check and tell them they would probably be heard by the inhabitants if they tried tunneling in.

Remember rule 0 and if your players are unable to accept that you are the ultimate authority at the table you might want to have a discussion with them about that.

That being said, try not to overly discourage creativity. It's fine to tell your players no but try not to do it too often.

Why not just put your encounters in locations that aren't underground?

Castles? Fortresses? Big towers? Mountains that already have extensive dug-out tunnels beneath the actual location of interest?

It's clear to you, and it should be clear to the Big Bad at this point, that underground is not a safe haven - so just don't give them that option.

My answer is similar to others, but should be different enough to merit a separate entry.

1. Top soil only goes down so far. Eventually you hit rock, and almost any underground fortress is going to be set in rock.

   Cave systems are left when water carves passages through rock and stone formations. The material left is hard enough to hold its shape around this void and is not going to be conducive to easy digging.

   Mine networks, especially the larger ones, are also usually fairly deep, and also mostly contained within rock formations (that's where the ore is).

2. In locations where that much digging by badgers is possible, it would be a prime territory for ankhegs and bulettes. That much digging will draw a lot of unwanted attention, even if the target of the tunneling remains unaware.

3. Sappers, when used in warfare to bypass or bring down walls, are only possible when the target is unaware, or when in a siege situation and the target is unable to attack, except with their own counter-sapper teams.