I catch accents like germs. I only have to listen to someone for a few minutes and I'm off speaking just like them. It's embarrassing sometimes, as I can't seem to stop doing it.

So, we have three choices:-

1) Try to imitate the original accent of the song.
2) Sing it in the accent of your own region.
3) Use a hybrid accent, a 'compromise' of the two.

I'd choose 1), but only if you can do it really convincingly.
If not, then 3).
It would be hilarious, in my opinion, to hear a Scots song sung in RP.
Or one of our Norfolk's Singing Postman's songs sung in a broad Welsh accent. It would be a comedy turn!