Catsidhe

I don't know of any nursery rhymes in Latin. I don't know that any are recorded from classical times, and by the middle ages Latin was not anyone's first language, and was taught long after the age that nursery rhymes would have been appropriate. There are, however, many recorded drinking songs in Latin. (Including gaudeamus igitur(!))

You may not, however, be entirely out of luck. Winnie ille Pu and Domus Anguli Puensis are translations of Winnie the Pooh and House at Pooh Corner into Latin. If the kids know these stories at all well, then the translations should be fairly transparent.

"Qui est?" rogavit Pu.
"Ego est," dixit animalis incognitum.
"Ah," dixit Pu.
...

If you can find them, of course. If you are in Melbourne, I'd try the Foreign Language Bookshop on Collins st first.