There is also a children's clapping game: Suzie had a baby, which may or may not be related.

Suzie had a baby, she called him Tiny Tim.
She put him in the bath-tub to see if he could swim.
He drank up all the water, he ate up all the soap.
He tried to eat the bath-tub but it wouldn't go down his throat.
Suzie called the doctor, Suzie called the nurse,
Suzie called the lady with the alligator purse.
In came the doctor, in came the nurse,
In came the lady with the alligator purse.
'Mumps' said the doctor, 'Measles' said the nurse,
'Chicken-pox' said the lady with the alligator purse.
Out went the doctor, out went the nurse,
Out went the lady with the alligator purse.

I think that's it.