I found a parody version HERE, but it does play the tune.

BTW Fred Heatherton's original opening verse appears to be: -

Down at the English Fair,
One ev'ning I was there,
When I heard a showman shouting
Underneath the blare.

The "Barnum" version seems to be a modification by various American performers - although the Danny Kaye words are the English ones. The lyrics sound much more likely to be English rather than American. Do you have coconut shies at American Fairs anyway ?