So firstly, my dialect is New Zealand English. Notable features are: non-rhotic, yod-retaining, mid-vowels are higher (so /æ ɛ/ are closer to [ɛ e]), the starting point of most diphthongs are lower, /ɪ/ is closer to [ɘ] and starting to merge with /ə/, and /ʌ/ is [ɐ] and contrasts with /ɐ:/ by length alone. Consonants are pretty standard.
So sound changes: stressed vowels before voiced consonants lengthen (I haven't decided whether this will destroy the old length contrast yet) and initial and final voiced consonants devoice, to contrast only in aspiration with the unvoiced consonants initially and in vowel length finally: cab, cap, gap → /kʰɛːp̚, kʰɛp̚, kɛp̚/. Intervocalic /t d/ → /ɾ/, intervocalic /b/ → /p/ and /g/ → /ɣ/ → /ŋ/. There is then a slight shuffling of the vowels, but nothing too far from modern-day NZ English, a reduction in consonant clusters (intervocalic /t/ comes back in a few instances, like arctic [ɐːtək̚]) and some l-vocalisation.
I'm not sure what a believable timeframe for this is, probably not too much more than 100 years though. It would still remain largely intelligible to speakers of other dialects of English, especially Australians, South Africans and some South England speakers.
Vowels
And just because it fits the vowels so well, I've tried using a variation of the Vietnamese alphabet to transcribe it, although realistically there probably wouldn't be any spelling reform.
Nyư Zealânt (ô Aotearoa, [phrâneonst Aôtêarrôa]) âz 'n ailânt khăntri ân dâ seođwestơn phâsâfâk ăưxân.
Maybe round 2's sound changes will be a merger of tr and ch, a fixing up of the vowels and semivowels and a neutralisation of voicing in the fricatives, maybe voicing intervocalically and devoicing word finally.