The one I learned VERY young via my mother and father from the long line of Aussie, Kiwi and Brit Royal Navy that streamed through our house on Long Island during WWII is most nearly like that posted by cranky yankee:

A mother was bathin' her bye-bee one night
Youngest of ten and a poor little mite.
The mother was poor and the bye-bee was thin
Just like a skeleton wrapped up in skin.

The mother turned round for the soap on the shelf
And when she turned back she was all by herself.
"Oh, where is my bye-bee in anguish she cried?"
And as she was sobbin', the angels replied:

"Your bye-bee is gone down the plug 'ole,
"Your bye-bee is gone down the plug.
"Poor little devil was awfully thin,
" 'T ought to 've been bathed in a jug.
"Your bye-bee is perfectly 'appy
"'Twon't need a bath any more.
"Your bye-bee is gone up to 'evvin
"Not lost, but just gone before.

Carter Bannerman