Eliminating Unicode accents in Apex

Someone has already published a complete solution. You just need to translate this answer from Javascript to Apex. The author of that answer credits this post, which was posted by lehel on May 6, 2011 and archived in 2012. I can't testify to the completeness, but here's the translation:

public class Accents
{
    public static String removeDiacritics(String text)
    {
        for (String letter : patterns.keySet())
            text = text.replaceAll(patterns.get(letter), letter);
        return text;
    }
    static Map<String, String> patterns = new Map<String, String>
    {
        'A' => '[\u0041\u24B6\uFF21\u00C0\u00C1\u00C2\u1EA6\u1EA4\u1EAA\u1EA8\u00C3\u0100\u0102\u1EB0\u1EAE\u1EB4\u1EB2\u0226\u01E0\u00C4\u01DE\u1EA2\u00C5\u01FA\u01CD\u0200\u0202\u1EA0\u1EAC\u1EB6\u1E00\u0104\u023A\u2C6F]',
        'AA' => '[\uA732]',
        'AE' => '[\u00C6\u01FC\u01E2]',
        'AO' => '[\uA734]',
        'AU' => '[\uA736]',
        'AV' => '[\uA738\uA73A]',
        'AY' => '[\uA73C]',
        'B' => '[\u0042\u24B7\uFF22\u1E02\u1E04\u1E06\u0243\u0182\u0181]',
        'C' => '[\u0043\u24B8\uFF23\u0106\u0108\u010A\u010C\u00C7\u1E08\u0187\u023B\uA73E]',
        'D' => '[\u0044\u24B9\uFF24\u1E0A\u010E\u1E0C\u1E10\u1E12\u1E0E\u0110\u018B\u018A\u0189\uA779]',
        'DZ' => '[\u01F1\u01C4]',
        'Dz' => '[\u01F2\u01C5]',
        'E' => '[\u0045\u24BA\uFF25\u00C8\u00C9\u00CA\u1EC0\u1EBE\u1EC4\u1EC2\u1EBC\u0112\u1E14\u1E16\u0114\u0116\u00CB\u1EBA\u011A\u0204\u0206\u1EB8\u1EC6\u0228\u1E1C\u0118\u1E18\u1E1A\u0190\u018E]',
        'F' => '[\u0046\u24BB\uFF26\u1E1E\u0191\uA77B]',
        'G' => '[\u0047\u24BC\uFF27\u01F4\u011C\u1E20\u011E\u0120\u01E6\u0122\u01E4\u0193\uA7A0\uA77D\uA77E]',
        'H' => '[\u0048\u24BD\uFF28\u0124\u1E22\u1E26\u021E\u1E24\u1E28\u1E2A\u0126\u2C67\u2C75\uA78D]',
        'I' => '[\u0049\u24BE\uFF29\u00CC\u00CD\u00CE\u0128\u012A\u012C\u0130\u00CF\u1E2E\u1EC8\u01CF\u0208\u020A\u1ECA\u012E\u1E2C\u0197]',
        'J' => '[\u004A\u24BF\uFF2A\u0134\u0248]',
        'K' => '[\u004B\u24C0\uFF2B\u1E30\u01E8\u1E32\u0136\u1E34\u0198\u2C69\uA740\uA742\uA744\uA7A2]',
        'L' => '[\u004C\u24C1\uFF2C\u013F\u0139\u013D\u1E36\u1E38\u013B\u1E3C\u1E3A\u0141\u023D\u2C62\u2C60\uA748\uA746\uA780]',
        'LJ' => '[\u01C7]',
        'Lj' => '[\u01C8]',
        'M' => '[\u004D\u24C2\uFF2D\u1E3E\u1E40\u1E42\u2C6E\u019C]',
        'N' => '[\u004E\u24C3\uFF2E\u01F8\u0143\u00D1\u1E44\u0147\u1E46\u0145\u1E4A\u1E48\u0220\u019D\uA790\uA7A4]',
        'NJ' => '[\u01CA]',
        'Nj' => '[\u01CB]',
        'O' => '[\u004F\u24C4\uFF2F\u00D2\u00D3\u00D4\u1ED2\u1ED0\u1ED6\u1ED4\u00D5\u1E4C\u022C\u1E4E\u014C\u1E50\u1E52\u014E\u022E\u0230\u00D6\u022A\u1ECE\u0150\u01D1\u020C\u020E\u01A0\u1EDC\u1EDA\u1EE0\u1EDE\u1EE2\u1ECC\u1ED8\u01EA\u01EC\u00D8\u01FE\u0186\u019F\uA74A\uA74C]',
        'OI' => '[\u01A2]',
        'OO' => '[\uA74E]',
        'OU' => '[\u0222]',
        'P' => '[\u0050\u24C5\uFF30\u1E54\u1E56\u01A4\u2C63\uA750\uA752\uA754]',
        'Q' => '[\u0051\u24C6\uFF31\uA756\uA758\u024A]',
        'R' => '[\u0052\u24C7\uFF32\u0154\u1E58\u0158\u0210\u0212\u1E5A\u1E5C\u0156\u1E5E\u024C\u2C64\uA75A\uA7A6\uA782]',
        'S' => '[\u0053\u24C8\uFF33\u1E9E\u015A\u1E64\u015C\u1E60\u0160\u1E66\u1E62\u1E68\u0218\u015E\u2C7E\uA7A8\uA784]',
        'T' => '[\u0054\u24C9\uFF34\u1E6A\u0164\u1E6C\u021A\u0162\u1E70\u1E6E\u0166\u01AC\u01AE\u023E\uA786]',
        'TZ' => '[\uA728]',
        'U' => '[\u0055\u24CA\uFF35\u00D9\u00DA\u00DB\u0168\u1E78\u016A\u1E7A\u016C\u00DC\u01DB\u01D7\u01D5\u01D9\u1EE6\u016E\u0170\u01D3\u0214\u0216\u01AF\u1EEA\u1EE8\u1EEE\u1EEC\u1EF0\u1EE4\u1E72\u0172\u1E76\u1E74\u0244]',
        'V' => '[\u0056\u24CB\uFF36\u1E7C\u1E7E\u01B2\uA75E\u0245]',
        'VY' => '[\uA760]',
        'W' => '[\u0057\u24CC\uFF37\u1E80\u1E82\u0174\u1E86\u1E84\u1E88\u2C72]',
        'X' => '[\u0058\u24CD\uFF38\u1E8A\u1E8C]',
        'Y' => '[\u0059\u24CE\uFF39\u1EF2\u00DD\u0176\u1EF8\u0232\u1E8E\u0178\u1EF6\u1EF4\u01B3\u024E\u1EFE]',
        'Z' => '[\u005A\u24CF\uFF3A\u0179\u1E90\u017B\u017D\u1E92\u1E94\u01B5\u0224\u2C7F\u2C6B\uA762]',
        'a' => '[\u0061\u24D0\uFF41\u1E9A\u00E0\u00E1\u00E2\u1EA7\u1EA5\u1EAB\u1EA9\u00E3\u0101\u0103\u1EB1\u1EAF\u1EB5\u1EB3\u0227\u01E1\u00E4\u01DF\u1EA3\u00E5\u01FB\u01CE\u0201\u0203\u1EA1\u1EAD\u1EB7\u1E01\u0105\u2C65\u0250]',
        'aa' => '[\uA733]',
        'ae' => '[\u00E6\u01FD\u01E3]',
        'ao' => '[\uA735]',
        'au' => '[\uA737]',
        'av' => '[\uA739\uA73B]',
        'ay' => '[\uA73D]',
        'b' => '[\u0062\u24D1\uFF42\u1E03\u1E05\u1E07\u0180\u0183\u0253]',
        'c' => '[\u0063\u24D2\uFF43\u0107\u0109\u010B\u010D\u00E7\u1E09\u0188\u023C\uA73F\u2184]',
        'd' => '[\u0064\u24D3\uFF44\u1E0B\u010F\u1E0D\u1E11\u1E13\u1E0F\u0111\u018C\u0256\u0257\uA77A]',
        'dz' => '[\u01F3\u01C6]',
        'e' => '[\u0065\u24D4\uFF45\u00E8\u00E9\u00EA\u1EC1\u1EBF\u1EC5\u1EC3\u1EBD\u0113\u1E15\u1E17\u0115\u0117\u00EB\u1EBB\u011B\u0205\u0207\u1EB9\u1EC7\u0229\u1E1D\u0119\u1E19\u1E1B\u0247\u025B\u01DD]',
        'f' => '[\u0066\u24D5\uFF46\u1E1F\u0192\uA77C]',
        'g' => '[\u0067\u24D6\uFF47\u01F5\u011D\u1E21\u011F\u0121\u01E7\u0123\u01E5\u0260\uA7A1\u1D79\uA77F]',
        'h' => '[\u0068\u24D7\uFF48\u0125\u1E23\u1E27\u021F\u1E25\u1E29\u1E2B\u1E96\u0127\u2C68\u2C76\u0265]',
        'hv' => '[\u0195]',
        'i' => '[\u0069\u24D8\uFF49\u00EC\u00ED\u00EE\u0129\u012B\u012D\u00EF\u1E2F\u1EC9\u01D0\u0209\u020B\u1ECB\u012F\u1E2D\u0268\u0131]',
        'j' => '[\u006A\u24D9\uFF4A\u0135\u01F0\u0249]',
        'k' => '[\u006B\u24DA\uFF4B\u1E31\u01E9\u1E33\u0137\u1E35\u0199\u2C6A\uA741\uA743\uA745\uA7A3]',
        'l' => '[\u006C\u24DB\uFF4C\u0140\u013A\u013E\u1E37\u1E39\u013C\u1E3D\u1E3B\u017F\u0142\u019A\u026B\u2C61\uA749\uA781\uA747]',
        'lj' => '[\u01C9]',
        'm' => '[\u006D\u24DC\uFF4D\u1E3F\u1E41\u1E43\u0271\u026F]',
        'n' => '[\u006E\u24DD\uFF4E\u01F9\u0144\u00F1\u1E45\u0148\u1E47\u0146\u1E4B\u1E49\u019E\u0272\u0149\uA791\uA7A5]',
        'nj' => '[\u01CC]',
        'o' => '[\u006F\u24DE\uFF4F\u00F2\u00F3\u00F4\u1ED3\u1ED1\u1ED7\u1ED5\u00F5\u1E4D\u022D\u1E4F\u014D\u1E51\u1E53\u014F\u022F\u0231\u00F6\u022B\u1ECF\u0151\u01D2\u020D\u020F\u01A1\u1EDD\u1EDB\u1EE1\u1EDF\u1EE3\u1ECD\u1ED9\u01EB\u01ED\u00F8\u01FF\u0254\uA74B\uA74D\u0275]',
        'oi' => '[\u01A3]',
        'ou' => '[\u0223]',
        'oo' => '[\uA74F]',
        'p' => '[\u0070\u24DF\uFF50\u1E55\u1E57\u01A5\u1D7D\uA751\uA753\uA755]',
        'q' => '[\u0071\u24E0\uFF51\u024B\uA757\uA759]',
        'r' => '[\u0072\u24E1\uFF52\u0155\u1E59\u0159\u0211\u0213\u1E5B\u1E5D\u0157\u1E5F\u024D\u027D\uA75B\uA7A7\uA783]',
        's' => '[\u0073\u24E2\uFF53\u00DF\u015B\u1E65\u015D\u1E61\u0161\u1E67\u1E63\u1E69\u0219\u015F\u023F\uA7A9\uA785\u1E9B]',
        't' => '[\u0074\u24E3\uFF54\u1E6B\u1E97\u0165\u1E6D\u021B\u0163\u1E71\u1E6F\u0167\u01AD\u0288\u2C66\uA787]',
        'tz' => '[\uA729]',
        'u' => '[\u0075\u24E4\uFF55\u00F9\u00FA\u00FB\u0169\u1E79\u016B\u1E7B\u016D\u00FC\u01DC\u01D8\u01D6\u01DA\u1EE7\u016F\u0171\u01D4\u0215\u0217\u01B0\u1EEB\u1EE9\u1EEF\u1EED\u1EF1\u1EE5\u1E73\u0173\u1E77\u1E75\u0289]',
        'v' => '[\u0076\u24E5\uFF56\u1E7D\u1E7F\u028B\uA75F\u028C]',
        'vy' => '[\uA761]',
        'w' => '[\u0077\u24E6\uFF57\u1E81\u1E83\u0175\u1E87\u1E85\u1E98\u1E89\u2C73]',
        'x' => '[\u0078\u24E7\uFF58\u1E8B\u1E8D]',
        'y' => '[\u0079\u24E8\uFF59\u1EF3\u00FD\u0177\u1EF9\u0233\u1E8F\u00FF\u1EF7\u1E99\u1EF5\u01B4\u024F\u1EFF]',
        'z' => '[\u007A\u24E9\uFF5A\u017A\u1E91\u017C\u017E\u1E93\u1E95\u01B6\u0225\u0240\u2C6C\uA763]'
    };
}

It sure does work with your desired input/output combo:

system.assertEquals('Hello', Accents.removeDiacritics('Ḧéļḻṏ'));

PS

This approach is painfully slow, taking approximately 1.8ms per execution. I would avoid using it in a loop where you may attempt to "clean" more than 1000 strings.

PPS

You can't compile this class in Execute Anonymous, but it will save in a top-level class. I guess there may be a limit to how many map keys you can specify in a literal constructor using Execute Anonymous, but I haven't isolated the cause of the error.

PPPS

As noted by Charles, you can take this concept and greatly increase performance by taking advantage of the getChars and fromCharArray methods. If you map each Unicode character catalogued below to the corresponding letter, it's about ten times faster.

This is a revision of Adrian Larson's previous answer for benchmark comparison:

static Map<Integer, String> sec = new Map<Integer, String>{
65=>'A', 9398=>'A', 65313=>'A', 192=>'A', 193=>'A', 194=>'A', 7846=>'A', 7844=>'A', 7850=>'A', 7848=>'A', 195=>'A', 256=>'A', 258=>'A', 7856=>'A', 7854=>'A', 7860=>'A', 7858=>'A', 550=>'A', 480=>'A', 196=>'A', 478=>'A', 7842=>'A', 197=>'A', 506=>'A', 461=>'A', 512=>'A', 514=>'A', 7840=>'A', 7852=>'A', 7862=>'A', 7680=>'A', 260=>'A', 570=>'A', 11375=>'A', 42802=>'AA', 198=>'AE', 508=>'AE', 482=>'AE', 42804=>'AO', 42806=>'AU', 42808=>'AV', 42810=>'AV', 42812=>'AY', 66=>'B', 9399=>'B', 65314=>'B', 7682=>'B', 7684=>'B', 7686=>'B', 579=>'B', 386=>'B', 385=>'B', 67=>'C', 9400=>'C', 65315=>'C', 262=>'C', 264=>'C', 266=>'C', 268=>'C', 199=>'C', 7688=>'C', 391=>'C', 571=>'C', 42814=>'C', 68=>'D', 9401=>'D', 65316=>'D', 7690=>'D', 270=>'D', 7692=>'D', 7696=>'D', 7698=>'D', 7694=>'D', 272=>'D', 395=>'D', 394=>'D', 393=>'D', 42873=>'D', 497=>'DZ', 452=>'DZ', 498=>'Dz', 453=>'Dz', 69=>'E', 9402=>'E', 65317=>'E', 200=>'E', 201=>'E', 202=>'E', 7872=>'E', 7870=>'E', 7876=>'E', 7874=>'E', 7868=>'E', 274=>'E', 7700=>'E', 7702=>'E', 276=>'E', 278=>'E', 203=>'E', 7866=>'E', 282=>'E', 516=>'E', 518=>'E', 7864=>'E', 7878=>'E', 552=>'E', 7708=>'E', 280=>'E', 7704=>'E', 7706=>'E', 400=>'E', 398=>'E', 70=>'F', 9403=>'F', 65318=>'F', 7710=>'F', 401=>'F', 42875=>'F', 71=>'G', 9404=>'G', 65319=>'G', 500=>'G', 284=>'G', 7712=>'G', 286=>'G', 288=>'G', 486=>'G', 290=>'G', 484=>'G', 403=>'G', 42912=>'G', 42877=>'G', 42878=>'G', 72=>'H', 9405=>'H', 65320=>'H', 292=>'H', 7714=>'H', 7718=>'H', 542=>'H', 7716=>'H', 7720=>'H', 7722=>'H', 294=>'H', 11367=>'H', 11381=>'H', 42893=>'H', 73=>'I', 9406=>'I', 65321=>'I', 204=>'I', 205=>'I', 206=>'I', 296=>'I', 298=>'I', 300=>'I', 304=>'I', 207=>'I', 7726=>'I', 7880=>'I', 463=>'I', 520=>'I', 522=>'I', 7882=>'I', 302=>'I', 7724=>'I', 407=>'I', 74=>'J', 9407=>'J', 65322=>'J', 308=>'J', 584=>'J', 75=>'K', 9408=>'K', 65323=>'K', 7728=>'K', 488=>'K', 7730=>'K', 310=>'K', 7732=>'K', 408=>'K', 11369=>'K', 42816=>'K', 42818=>'K', 42820=>'K', 42914=>'K', 76=>'L', 9409=>'L', 65324=>'L', 319=>'L', 313=>'L', 317=>'L', 7734=>'L', 7736=>'L', 315=>'L', 7740=>'L', 7738=>'L', 321=>'L', 573=>'L', 11362=>'L', 11360=>'L', 42824=>'L', 42822=>'L', 42880=>'L', 455=>'LJ', 456=>'Lj', 77=>'M', 9410=>'M', 65325=>'M', 7742=>'M', 7744=>'M', 7746=>'M', 11374=>'M', 412=>'M', 78=>'N', 9411=>'N', 65326=>'N', 504=>'N', 323=>'N', 209=>'N', 7748=>'N', 327=>'N', 7750=>'N', 325=>'N', 7754=>'N', 7752=>'N', 544=>'N', 413=>'N', 42896=>'N', 42916=>'N', 458=>'NJ', 459=>'Nj', 79=>'O', 9412=>'O', 65327=>'O', 210=>'O', 211=>'O', 212=>'O', 7890=>'O', 7888=>'O', 7894=>'O', 7892=>'O', 213=>'O', 7756=>'O', 556=>'O', 7758=>'O', 332=>'O', 7760=>'O', 7762=>'O', 334=>'O', 558=>'O', 560=>'O', 214=>'O', 554=>'O', 7886=>'O', 336=>'O', 465=>'O', 524=>'O', 526=>'O', 416=>'O', 7900=>'O', 7898=>'O', 7904=>'O', 7902=>'O', 7906=>'O', 7884=>'O', 7896=>'O', 490=>'O', 492=>'O', 216=>'O', 510=>'O', 390=>'O', 415=>'O', 42826=>'O', 42828=>'O', 418=>'OI', 42830=>'OO', 546=>'OU', 80=>'P', 9413=>'P', 65328=>'P', 7764=>'P', 7766=>'P', 420=>'P', 11363=>'P', 42832=>'P', 42834=>'P', 42836=>'P', 81=>'Q', 9414=>'Q', 65329=>'Q', 42838=>'Q', 42840=>'Q', 586=>'Q', 82=>'R', 9415=>'R', 65330=>'R', 340=>'R', 7768=>'R', 344=>'R', 528=>'R', 530=>'R', 7770=>'R', 7772=>'R', 342=>'R', 7774=>'R', 588=>'R', 11364=>'R', 42842=>'R', 42918=>'R', 42882=>'R', 83=>'S', 9416=>'S', 65331=>'S', 7838=>'S', 346=>'S', 7780=>'S', 348=>'S', 7776=>'S', 352=>'S', 7782=>'S', 7778=>'S', 7784=>'S', 536=>'S', 350=>'S', 11390=>'S', 42920=>'S', 42884=>'S', 84=>'T', 9417=>'T', 65332=>'T', 7786=>'T', 356=>'T', 7788=>'T', 538=>'T', 354=>'T', 7792=>'T', 7790=>'T', 358=>'T', 428=>'T', 430=>'T', 574=>'T', 42886=>'T', 42792=>'TZ', 85=>'U', 9418=>'U', 65333=>'U', 217=>'U', 218=>'U', 219=>'U', 360=>'U', 7800=>'U', 362=>'U', 7802=>'U', 364=>'U', 220=>'U', 475=>'U', 471=>'U', 469=>'U', 473=>'U', 7910=>'U', 366=>'U', 368=>'U', 467=>'U', 532=>'U', 534=>'U', 431=>'U', 7914=>'U', 7912=>'U', 7918=>'U', 7916=>'U', 7920=>'U', 7908=>'U', 7794=>'U', 370=>'U', 7798=>'U', 7796=>'U', 580=>'U', 86=>'V', 9419=>'V', 65334=>'V', 7804=>'V', 7806=>'V', 434=>'V', 42846=>'V', 581=>'V', 42848=>'VY', 87=>'W', 9420=>'W', 65335=>'W', 7808=>'W', 7810=>'W', 372=>'W', 7814=>'W', 7812=>'W', 7816=>'W', 11378=>'W', 88=>'X', 9421=>'X', 65336=>'X', 7818=>'X', 7820=>'X', 89=>'Y', 9422=>'Y', 65337=>'Y', 7922=>'Y', 221=>'Y', 374=>'Y', 7928=>'Y', 562=>'Y', 7822=>'Y', 376=>'Y', 7926=>'Y', 7924=>'Y', 435=>'Y', 590=>'Y', 7934=>'Y', 90=>'Z', 9423=>'Z', 65338=>'Z', 377=>'Z', 7824=>'Z', 379=>'Z', 381=>'Z', 7826=>'Z', 7828=>'Z', 437=>'Z', 548=>'Z', 11391=>'Z', 11371=>'Z', 42850=>'Z', 97=>'a', 9424=>'a', 65345=>'a', 7834=>'a', 224=>'a', 225=>'a', 226=>'a', 7847=>'a', 7845=>'a', 7851=>'a', 7849=>'a', 227=>'a', 257=>'a', 259=>'a', 7857=>'a', 7855=>'a', 7861=>'a', 7859=>'a', 551=>'a', 481=>'a', 228=>'a', 479=>'a', 7843=>'a', 229=>'a', 507=>'a', 462=>'a', 513=>'a', 515=>'a', 7841=>'a', 7853=>'a', 7863=>'a', 7681=>'a', 261=>'a', 11365=>'a', 592=>'a', 42803=>'aa', 230=>'ae', 509=>'ae', 483=>'ae', 42805=>'ao', 42807=>'au', 42809=>'av', 42811=>'av', 42813=>'ay', 98=>'b', 9425=>'b', 65346=>'b', 7683=>'b', 7685=>'b', 7687=>'b', 384=>'b', 387=>'b', 595=>'b', 99=>'c', 9426=>'c', 65347=>'c', 263=>'c', 265=>'c', 267=>'c', 269=>'c', 231=>'c', 7689=>'c', 392=>'c', 572=>'c', 42815=>'c', 8580=>'c', 100=>'d', 9427=>'d', 65348=>'d', 7691=>'d', 271=>'d', 7693=>'d', 7697=>'d', 7699=>'d', 7695=>'d', 273=>'d', 396=>'d', 598=>'d', 599=>'d', 42874=>'d', 499=>'dz', 454=>'dz', 101=>'e', 9428=>'e', 65349=>'e', 232=>'e', 233=>'e', 234=>'e', 7873=>'e', 7871=>'e', 7877=>'e', 7875=>'e', 7869=>'e', 275=>'e', 7701=>'e', 7703=>'e', 277=>'e', 279=>'e', 235=>'e', 7867=>'e', 283=>'e', 517=>'e', 519=>'e', 7865=>'e', 7879=>'e', 553=>'e', 7709=>'e', 281=>'e', 7705=>'e', 7707=>'e', 583=>'e', 603=>'e', 477=>'e', 102=>'f', 9429=>'f', 65350=>'f', 7711=>'f', 402=>'f', 42876=>'f', 103=>'g', 9430=>'g', 65351=>'g', 501=>'g', 285=>'g', 7713=>'g', 287=>'g', 289=>'g', 487=>'g', 291=>'g', 485=>'g', 608=>'g', 42913=>'g', 7545=>'g', 42879=>'g', 104=>'h', 9431=>'h', 65352=>'h', 293=>'h', 7715=>'h', 7719=>'h', 543=>'h', 7717=>'h', 7721=>'h', 7723=>'h', 7830=>'h', 295=>'h', 11368=>'h', 11382=>'h', 613=>'h', 405=>'hv', 105=>'i', 9432=>'i', 65353=>'i', 236=>'i', 237=>'i', 238=>'i', 297=>'i', 299=>'i', 301=>'i', 239=>'i', 7727=>'i', 7881=>'i', 464=>'i', 521=>'i', 523=>'i', 7883=>'i', 303=>'i', 7725=>'i', 616=>'i', 305=>'i', 106=>'j', 9433=>'j', 65354=>'j', 309=>'j', 496=>'j', 585=>'j', 107=>'k', 9434=>'k', 65355=>'k', 7729=>'k', 489=>'k', 7731=>'k', 311=>'k', 7733=>'k', 409=>'k', 11370=>'k', 42817=>'k', 42819=>'k', 42821=>'k', 42915=>'k', 108=>'l', 9435=>'l', 65356=>'l', 320=>'l', 314=>'l', 318=>'l', 7735=>'l', 7737=>'l', 316=>'l', 7741=>'l', 7739=>'l', 383=>'l', 322=>'l', 410=>'l', 619=>'l', 11361=>'l', 42825=>'l', 42881=>'l', 42823=>'l', 457=>'lj', 109=>'m', 9436=>'m', 65357=>'m', 7743=>'m', 7745=>'m', 7747=>'m', 625=>'m', 623=>'m', 110=>'n', 9437=>'n', 65358=>'n', 505=>'n', 324=>'n', 241=>'n', 7749=>'n', 328=>'n', 7751=>'n', 326=>'n', 7755=>'n', 7753=>'n', 414=>'n', 626=>'n', 329=>'n', 42897=>'n', 42917=>'n', 460=>'nj', 111=>'o', 9438=>'o', 65359=>'o', 242=>'o', 243=>'o', 244=>'o', 7891=>'o', 7889=>'o', 7895=>'o', 7893=>'o', 245=>'o', 7757=>'o', 557=>'o', 7759=>'o', 333=>'o', 7761=>'o', 7763=>'o', 335=>'o', 559=>'o', 561=>'o', 246=>'o', 555=>'o', 7887=>'o', 337=>'o', 466=>'o', 525=>'o', 527=>'o', 417=>'o', 7901=>'o', 7899=>'o', 7905=>'o', 7903=>'o', 7907=>'o', 7885=>'o', 7897=>'o', 491=>'o', 493=>'o', 248=>'o', 511=>'o', 596=>'o', 42827=>'o', 42829=>'o', 629=>'o', 419=>'oi', 547=>'ou', 42831=>'oo', 112=>'p', 9439=>'p', 65360=>'p', 7765=>'p', 7767=>'p', 421=>'p', 7549=>'p', 42833=>'p', 42835=>'p', 42837=>'p', 113=>'q', 9440=>'q', 65361=>'q', 587=>'q', 42839=>'q', 42841=>'q', 114=>'r', 9441=>'r', 65362=>'r', 341=>'r', 7769=>'r', 345=>'r', 529=>'r', 531=>'r', 7771=>'r', 7773=>'r', 343=>'r', 7775=>'r', 589=>'r', 637=>'r', 42843=>'r', 42919=>'r', 42883=>'r', 115=>'s', 9442=>'s', 65363=>'s', 223=>'s', 347=>'s', 7781=>'s', 349=>'s', 7777=>'s', 353=>'s', 7783=>'s', 7779=>'s', 7785=>'s', 537=>'s', 351=>'s', 575=>'s', 42921=>'s', 42885=>'s', 7835=>'s', 116=>'t', 9443=>'t', 65364=>'t', 7787=>'t', 7831=>'t', 357=>'t', 7789=>'t', 539=>'t', 355=>'t', 7793=>'t', 7791=>'t', 359=>'t', 429=>'t', 648=>'t', 11366=>'t', 42887=>'t', 42793=>'tz', 117=>'u', 9444=>'u', 65365=>'u', 249=>'u', 250=>'u', 251=>'u', 361=>'u', 7801=>'u', 363=>'u', 7803=>'u', 365=>'u', 252=>'u', 476=>'u', 472=>'u', 470=>'u', 474=>'u', 7911=>'u', 367=>'u', 369=>'u', 468=>'u', 533=>'u', 535=>'u', 432=>'u', 7915=>'u', 7913=>'u', 7919=>'u', 7917=>'u', 7921=>'u', 7909=>'u', 7795=>'u', 371=>'u', 7799=>'u', 7797=>'u', 649=>'u', 118=>'v', 9445=>'v', 65366=>'v', 7805=>'v', 7807=>'v', 651=>'v', 42847=>'v', 652=>'v', 42849=>'vy', 119=>'w', 9446=>'w', 65367=>'w', 7809=>'w', 7811=>'w', 373=>'w', 7815=>'w', 7813=>'w', 7832=>'w', 7817=>'w', 11379=>'w', 120=>'x', 9447=>'x', 65368=>'x', 7819=>'x', 7821=>'x', 121=>'y', 9448=>'y', 65369=>'y', 7923=>'y', 253=>'y', 375=>'y', 7929=>'y', 563=>'y', 7823=>'y', 255=>'y', 7927=>'y', 7833=>'y', 7925=>'y', 436=>'y', 591=>'y', 7935=>'y', 122=>'z', 9449=>'z', 65370=>'z', 378=>'z', 7825=>'z', 380=>'z', 382=>'z', 7827=>'z', 7829=>'z', 438=>'z', 549=>'z', 576=>'z', 11372=>'z', 42851=>'z' 
  };

String removeDiacritics(String str) {
  List<Integer> result = new List<Integer>();
  for (Integer chr: str.getChars()) {
    if (sec.containsKey(chr)) {
      result.addAll(sec.get(chr).getChars());
    } else {
      result.add(chr);
    }
  }
  return String.fromCharArray(result);
}

There's no simple transform method built-in to the library. If you wanted to take the time to map them all out, it would be possible to use a long series of String.replaceAll to manually go through the process, something like:

myString = myString.replaceAll('[ÀÁÂÃÄÅ]','A');
myString = myString.replaceAll('[àáâãäå]','a');
// ....

This would undoubtedly be particularly painful, but it could be possible.