My accent is from New York City, yet I wonder which area has the most or least sounds in their phonemic inventory. While one may have the most vowels and another the most consonants, I would like to hear that as well.
I think you are asking about the greatest number of phonemes.
I don't know which variety of English has the most phonemes, but my guess is that it may well be so-called Standard British (also called Southern Standard British English, or Recieved Pronunciation, RP).
There are a several reasons for thinking this:
It is non-rhotic, meaning that there is no /r/ in syllable codas. This in turn means that it has several vowels which only occur in environments where syllable final /r/ has been lost.
It was subject to vowel splits such as the TRAP/BATH split, which do not occur in many other regional varieties of English.
It has not undergone vowel mergers such as the Mary/marry/merry merger, which have occurred in other Englishes.
It currently retains distinction between voiced th, /ð/, and /v, d/. It also retains the distinction between voiceless th, /θ/, and /f, t/. Both of these have been lost in some British and other Englishes.
Nearly all the differences in numbers of phonemes—that I know of— between Standard British and other varieties involve Standard British having more distinct phonemes than its cousins.
I have no idea about which variety or varieties have the least phonemes. However, I would guess that it is a rhotic variety which has lost its dental fricatives, /θ/ and /ð/, and which has both been subject to various vowel mergers and not experienced many of the vowel splits seen in other varieties. It is unlikely that there will be any significant difference in the inventory of consonant phonemes bewteen different varieties.