There are several responses on http://www.answerbag.com/q_view/450329 to the question:

"Speaking of the name Mary, Does anyone know the nursery rhyme that went 'Aunty Mary had a Canary, she also had a duck?"

Here's one response that included some hyperlinked citations. I'm reposting it here exactly as it is found on that website:

by VSPrasad48 on Nov 2, 2007 at 8:17 pm

Aunt Mary had a canary
Up the leg of her drawers
She pulled the string to hear it sing
And down came Santa Claus

Uncle Jock, he had a sock
Up the pleat of his kilt
When he was a-sleepin, we were a-peepin
To see how well he was built

Chase me Charlie, I've got barley
Up the leg of me drawers
If you don't believe me, come and feel me
Up the leg of me drawers

Cousin Minnie wore a bikini
Underneath her shirt
A handsome guy he tried to spy
And she kicked him where it hurt.

http://www.footstompin.com/forum

http://www.sausagenet.org/nph-YaBB.pl

One fine day, a bonnie bonnie day,
I met my Auntie Mary;
She gied me a penny,
Tae buy whit I liked,
So I bocht a wee canary.
The wee canary flew awa'
Auntie Mary caught it;
She caught it by the tail
An' flung it in a pail,
Cheeky Auntie Mary.

My mother said that I must go,
Wi ma faither's denner-o;
Chappit tatties, beef and steak,
Twa reed herrin an' a bawbee bake.

I went to the river an' a couldn't get across,
I paid ten pound for an auld dun horse.
I jumped on his back an' his banes gied a crack,
An' I had tae play the fiddle till the boat came back.

http://www.footstompin.com/forum

Vaudeville (USA) -(UK) musichall became popular with a wider variety of entertainment & the minstrel shows began to wane although one of the variety acts would often be a comedy quartet. Songs both popular, classical and religion were parodied in keeping with events of the day not just by the singing acts but by the comedians too as short 1 or 2-line musical jokes.

Oh we've got chickens in our backyard & we feed them on cod oil (repeat)
But Aunty Mary's got a canary - he's the cock of the north (repeat)

(Scottish traditional melody)

http://labbs.org.uk/educ/resources/roots.html