I have the book; both the words and the tune are slightly different from what's posted above. I'll type the words when I have time.

I learned a very different version, The Sack 'em up Boys" in a Scots Music Group song class: the only part that's the same is the chorus (near enough) but the rest of the words and the tune are totally different. We were told that the chorus was an old Edinburgh street song, and that possibly the words were supplied by Robin Hall and Jimmy MacGregor. I'll sing it for you in a Zoom session...

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.....
C=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