Several of my (Scottish) songwriting friends do write in Americana style, and therefore sing in a faux American accent, which is maybe fair enough, tho' I guess our American friends would not identify it as being from any defined area of their great country, if even passing for American at all.
I would go with Anne's suggestion of keeping fairly neutral otherwise, and letting the words of the song speak for themselves, without a curious false accent becoming a distraction.
Having lived in the East End of London during my student days and beyond, for 7 years, I have a rough idea of what a Cockney accent sounds like, and genuine Cockneys are supposed to have been born and brought up "within the sound of Bow Bells": many London accents (and there ARE many to the well-tuned ear) are therefore not Cockney, so poor old Dick V-D didn't 'ave a cat's chance in 'ell, did 'e, of gettin' it raht?
(And there are many very local accents in other parts of the UK: I'm always surprised when locally raised friends say something like - "That's no' Broxburn, that's Whitburn " - the two places being but a few miles apart.)