I learned this song on the bus on the way to a high school debate tournament. You have to sing it really, really fast.

The Donut Song
Well, I walked around the corner
And I walked around the block
And I walked right into a bakery shop
And I picked up a donut
And I wiped off the grease
And I handed the lady a five-cent piece.

Well, she looked at the nickel and she looked at me
And she said "Hey mister, can't you plainly see?
There's a hole in the nickel,
there's a hole right through!"
I said, "There's a hole in the donut too."
"Thanks for the donut. Goodbye!"

This isn't really a hand jive because there are no accompanying motions (as far as I know) and it isn't really a nursery rhyme. It's just an annoying song.