The one I knew as a kiddie growing up in the late 80s was similar to Bec's version:

Aunty Mary had a canary
She also had a duck
She took them behind the kitchen door
And taught them how to...

Fry an egg and bacon
And brew a cup of tea
The more you eat, the more you drink
the more you want to...

Peter had a row boat
The boat began to rock
A shark came up the starboard bow
And bit him on the...

Cockle are great for fishing
Mary's in the punt(Not 100%)
The fishing line got tangled
And caught on Mary's...

Country girls are beautiful
Lying in the grass
They stick their heads between their legs
And whistle up their...

Ask your mum for sixpence
To see the tall giraffe
With buttons on his breast coat
And pimples up his...

Aunty Mary had a canary... (repeat from top)


Or something akin to that...