Only 20 years later, but referring back to the posts by Rapparee and Suzanne on 20th January 2003, here is the song that Rapparee referred to (which as Suzanne says, is not the same one as Robin Laing's song "Burke and Hare". I learned it in one of my singing classes in Edinburgh. We were told it did come from an old street song, as Rapparee suggested, which became the chorus, then possibly Robin Hall and Jimmy McGregor added some verses to it. I don't know of any recordings of it, but could write out the tune if Suzanne is still looking for it!
Incidentally, the Dr Knox has nothing to do with John Knox, the preacher, but refers to Robert Knox, the then anatomy demonstrator at the Edinburgh Medical School, who required bodies to teach his students.

THE SACK ‘EM UP BOYS
Chorus
Up the close and doon the stair,
But and ben wi’ Burke and Hare,
Burke’s the butcher,
Hare’s the thief,
Knox the boy who buys the beef.

Hurry doon the Castle Wynd,
Look before and look behind,
There they wait tae tak yer life
And sell ye fur the surgeon’s knife....
Chorus

Auld or young or dark or fair,
It maks na mind tae Burke and Hare,
While Dr Knox peys oot the tin,
They’ll sack ‘em up and bring them in.....
Chorus

Reekie’s rows are dark and drear,
Reekie’s vennels reek wi’ fear,
Mind yersel gaun doon the stair,
Fur fear ye meet wi’ Burke and Hare.....
Chorus