Bill the sound:
Ed Miller also recorded 'Burke and Hare' to the tune of 'The Fox and the Hare' with words written by Robin Laing; it is on 'Live at the Cactus Cafe.' The chorus is

Burke and Hare were a terrible pair,
Their deeds were beyond belief.
They worked underground in Edinburgh town,
The cruelest kind of thief.

Is that the one you were seeking?
He also sings 'The Edinburgh Rambler' which is Ewan MacColl's 'The Manchester Rambler' with lyrics altered to describe the border hills.
Anyone looking for Edinburgh songs would do well to seek out Ed's work; his website is
Songs of Scotland
Also, Greentrax' Celtic Collection Series Volume 13 is 'The Music and Song of Edinburgh' by various artists, all wonderful. Most of the songs mentioned above are on it. Their site is
Greentrax