This might be fine for classical, but it doesn't work for rock or pop, nor folk either. So I think that's a major factor for why British people often end up with a slight American accent when they sing

I never, ever, ever sing with a slight American accent, ever. Not ever. Sometimes when I learn a song from a record I find I've picked up a bit of the accent it was sung in (Paddy McAloon's faint Irish is good, or bad, for this) and I have to school myself to sing in my own voice. But American? Never. (And I have sung bits of Dylan, although that's something of a special case - an off-the-peg 'mid-Atlantic' accent wouldn't sound anything like him, but if you did try and sound him it would sound either spooky* or ridiculous.)

Dunno about the elongated vowels - my sense is that a lot of the long notes in traditional songs have long vowels to go with them ("the trees they do grow hiiiigh and the leaves they do grow greeeen"). It is an oral tradition, after all - the songs wouldn't have come down to us in the shape they're in if they were unsingable, or unsingable without sounding posh or American.

*I did once hear a young bloke doing Dylan numbers who sounded *exactly* like the man himself, right down to the picking style - and it was a bit spooky. Wonder if he's made a career out of it.