I can recall excerpts rom two songs which date (at least) from the early 1960's. The Davy Crockett reference was presumably from a TV series shown on TV after school in those days. Funnily enough, I don't ever remember hearing more than thatr first couplet:

"Say what you will, school dinners make you ill
Davy Crockett died of Shepherd's pie!"
[followed by laughter f course.]

The other is contemporaneous with the first (so far as I know) - although of course these things may go back generations, just as the playground games do...

"Ugly, mugly custard, bogey snotty pie
All mixed up in a dead dog's eye
Stir it, stir it, stir it thick
Was it all down with a hot cup of sick!"

LOL!!