The Auld Toon Shuffle made me clean forget my original question:

Can anyone point me to a recording of the song on Burke and Hare with the chorus (more or less)
Up the close and doon the stair
Through the town with Burke and Hare

I once heard Nancy Nicolson sing it, but she doesn't seem to have recorded it. The song on Robin Laing's 'Edinburgh Skyline' is a self-penned one - good, but not what I'm looking for.