Many moons ago, I sang at a musical evening for fellow teachers, a selection of folk music including "Aupres de ma Blonde", which I paired with "This one and that one may court me", both being about loves missing in the Netherlands at war. My version of the French came from my father's French teacher when he was in the army during WW2, at Clapham. It transpired that there were two French teachers in the audience, and I was a bit embarrassed, but they said it was OK (apart from one grammatical slip involving tous and tout), so it remains in my repertoire. I do think that singing in the original language is part of respecting the source. (Though providing explanation somewhere is a good idea.) Dialect, that is alternative vocabulary and syntax, I feel, should receive the same respect. Very often, the way the words work with the tune can be lost by changing the dialect to something different (I'm thinking of the way some modern hymn editors change words from Victorian hymn dialect to modern forms.)
Accent is different. I am one of those who picks accents up easily - currently, to my shame, it's a bit estuarine, innit. I do not feel comfortable singing innit. My singing is much more received. I don't make any attempt to replicate accents when singing - it doesn't come nearly as easily as spoken accents do, and would interfere with the singing process, I think. The conflict between the mouth shape needed for the singing sound and that needed for the accent is probably what leads to the forced effect.
I have been made aware at times that people with accents unlike mine regard the subject as sensitive. I was banned from reading "Albert and the Lion" to the children by a Mancunian colleague. A friend's wife was deeply offended by my saying that she had a Yorkshire accent - according to her, she didn't have any at all, but she did, though weak. I use them when reading to children, when they are relevant (ie, when they distinguish characters from each other), but only when I have heard the accent used recently. I certainly wouldn't do it in front of people I don't know, because it's a minefield. I do have problems in the literacy hour with "poems from different cultures" by Agard and Zephaniah, for example, because I can only do Trinidadian, and it's not up to date, and I don't know the writers' views on how their work should be read by estuarine white teachers. And their poems are impossible to read with respect without some indication of the natural rhythms and tones of the writers' language. As is Albert. A difficult area all round, not just with singing.

Penny