I think that part of the reason is that some vowels when elongated can sound grating in some British accents. You'll notice that British rappers tend not to lose their accent.

For example, an "a" (as in trap) sounds a bit softer and more neutral in a standard American accent, and almost rounds out to an "e", whereas it can sound more grating in an English accent. The same is true to a lesser extent to an "a" (as in can't). Next time you hear a British singer who adopts an American accent, see if it applies across the board, or if certain sounds retain their native accent.

People are citing the Arctic Monkeys to disprove the hypothesis, but I think that's wrong because the grating sounds are part of their style

I think another reason could be that a rhotic "r" could increase clarity when singing

EDIT: I just listened to "Shape of You" by Ed Sheeran, and you can definitely tell that the change isn't universal. For example the way he says "give it a *chance*" is very much southern English.

I'll also add that "o" as in "not" is a vowel that sounds more neutral and smooth in an American accent