Joe Offer- hello! You've been about as scarce as I have, here. Welcome back. The version used by Cindy and Ann seems to be the one I have sung much in the past. At least the first two verses are. The use of the word "Bandyrowe" is mine, and the verse about "little Jon" is mine...when I was rocking my little boys, I made up a verse for each:

Next come in was little Jon
Kitty alone, kitty alone,
Next come in was little Jon
Kitty alone-alye,
Next come in was little Jon,
One shoe off and one shoe on!
Kitty alone-alye, rocka-ma-rye-ree.

Next come in was little Pete
Kitty alone, kitty alone,
Next come in was little Pete
Kitty alone-alye,
Next come in was little Pete,
Fixin around to go to sleep!
Kitty alone-alye, rocka-ma-rye-ree.

"Fixin around to go to sleep" refers back to an earlier verse about two little ants, "Fixin around to have a dance," so Pete's verse would end the song.