Rwan, you'll have seen the version in Barke's essay on "pornography & Bawdry in Literature and Society" [a very pompous title, that!] in the Smith-Barke-Ferguson edition of The Merry Muses of Caledonia (1959 etc.), which is the same as Bob's above. The "Barnum & Bailey" version is in Nicht at Eenie (1932), 33, with music, whence Montgomerie, Scottish Nursery Rhymes (1946), 95.[This has no mention of unmentionables.] Nicht at Eenie has another, ibid., which may be the original of that one, namely

"Sister Mary had a canary
Whustled "The Cock o' the North."
It whustled for hoors & frightened the Booers,
And won the Victoria Cross.

A long time since I heard just a fragment of what may be another version,:-

"Auntie Mary had a canary
Up the leg o' her breeks"
[Leslie, Fife, circa 1940 maybe].

But I don't see what the ensuing rhyme would be.