Pyth - 70 68 bytes

K^T5J^T3W>KZ%." r3úBTê;¥
í".D/KJ60=J?<OT3O^T4J=-KJ.d1.

Try it online without sleeps.

MATL, 78 bytes

Thanks to @Maltysen and @DigitalTrauma for corrections.

1e5 1e3`XK10&XxyXIy/t0>*12L/'MM:SS'XO'Time remaining: 'whD-r.3<?1e4Yr}K]I0>]xx

Try it at MATL Online! (you may need to press "Run" several times if it doesn't initially work). Note that the online interpretet times out after 30 seconds, so the program may not have time to finish. You may want to change 10 (pause time in tenths of second) to something smaller such as 3 in order to increase speed of display

Explanation

1e5                  % Push 1e5: file size in 10-kB units
1e3                  % Push 1e3: initial rate in 10-kB/s units
`                    % Do...while
  XK                 %   Copy current trate into clipboard K
  10&Xx              %   Wait 1 second and clear screen
  y                  %   Duplicate current file size onto the top of the stack
  XI                 %   Copy it to clipboard I
  y                  %   Duplicate current rate onto the top of the stack
  /                  %   Divide. This gives the estimated remaining time in seconds
                     %   It may be negative in the last iteration, because the
                     %   "remaining" file size may have become negative
  t0>*               %   If negative, convert to 0
  12L/               %   Push 86400 and divide, to convert from seconds to days
  'MM:SS'XO          %   Format as a MM:SS string, rounding down
  'Time remaining: ' %   Push this string
  wh                 %   Swap, concatenate
  D                  %   Display
  -                  %   Subtract. This gives the new remaining file size
  r                  %   Push random number uniformly distributed in (0,1)
  .3<                %   Is it less than 0.3?
  ?                  %   If so
    1e4Yr            %     Random integer between 1 and 1e4. This is the new rate 
                     %     in 10-kB/s units
  }                  %   Else
    K                %     Push rate that was copied into clipboard K
  ]                  %   End
  I                  %   Push previous remaining file size from clipboard I
  0>                 %   Is it positive?
]                    % End. If top of the stack is true: next iteration
xx                   % Delete the two numbers that are on the stack

PowerShell, 190 bytes

$f=1gb;$r=10mb;while($f-gt0){if((1..10|Random)-le3){$r=(Random -mi 1kb -ma (10mb+1))*10}"Time remaining: "+((get-date 2/1/00).addSeconds($f/$r)-(get-date 2/1/00))-f'hh:mm:SS';$f-=$r;sleep 1}

Try it online! (TIO doesn't support clearing screen between lines)

Sets our initial $file size and our initial transfer $rate to 1gb and 10mb/s, respectively. Then, so long as we still have $file remaining, we loop.

Inside the loop, the if selects a number from 1 to 10 inclusive, and if it's 1, 2, or 3 (i.e., 30% of the time), we change the rate. This picks a random integer between 1kb and 10mb then that's multiplied by 10 to get our step count.

We then leverage a quirk of the Get-Date function, in that when given just a date, it returns a datetime object at 00:00:00am. This allows us to add the appropriate amount of seconds (based on the current rate and how much time is left), then subtract that date again, so we have a datetime object of how many seconds remain. That we can feed through the -format function, so it outputs in the desired format. Note we're not doing any rounding of seconds, and this includes leading hours, even if empty (pending decision from OP).

Ruby, 116 bytes

Try it online, except repl.it can't actually use Unix clear so it prints line by line instead.

f=1e5;r=1e3;(k=f/r=rand<0.3?1+rand(1e4):r;puts`clear`+"Time remaining: %02d:%02d"%[k/60,k%60];f-=r;sleep 1)while f>0

Windows version takes +1 byte by replacing the puts call with puts"\e[H\e[2JTime remaining: %02d:%02d"%[k/60,k%60] instead.

Bash + common utils, 117

Straightforward implementation. A few bytes saved by dividing out by 10000:

for((b=10**5,r=1000;b>0;r=RANDOM%10<3?RANDOM%10000+1:r,b-=r));{
clear
date -ud@$[b/r] "+Time remaining: %T"
sleep 1
}

Try it online. Using sleep 0 on the TIO so you don't have to wait. clear doesn't work on TIO.