Damn - tag fail. Let's try again.

I wouldn't dream of anglicising a song like that - it wouldn't seem like 'anglicising', in fact, it would seem like translating it from Scots into English.

Perhaps we can distinguish between songs that are in Scots and those that are in English with some Scots vocabulary & word forms? Take Child's Sheath and Knife, for example (variant A):

He's taen his sister doun to her father's deer park,
Wi his yew-tree bow and arrows fast slung to his back.

'Now when that ye hear me gie a loud cry,
Shoot frae thy bow an arrow and there let me lye.

It looks recognisably 'Scottish', but it drops into standard English like shelling peas - you hardly need to change anything. A song like Twa Corbies, on the other hand -

As I was walking all alane,
I heard twa corbies makin a mane;
The tane unto the ither say,
"Whar sall we gang and dine the-day?"

- that's Scots. As an English singer you either sing it as it's written and take your chances with the accent, or you translate it (alone, two, moan, one, other, where, shall, go, today, and so on through the rest of the song). Or leave it alone, of course.

Having said that, I think I could probably do a passable impression of a Scot singing that song, just as I can do a reasonable impression of a Yorkshireman singing "Old Molly Metcalfe": when you learn a song from a single source, the accent and intonation of the singer sticks with you. But there would always be a temptation to "do" an accent, and (unless you're an impressionist) the accents you consciously put on are rarely recognisable to anyone else.