Here's the version I learned as a child. Clearly much mutated from but also related to other versions:

In Chinatown there lived a man
His name was Chickety-Chickety Chan
His feet were long and his toes were short
And this is the way the Chinamen talk

CHORUS:
Ooooh, chickety-chee chi-ly chi-lo
chickety-rummity inna-ga-waddiya
waddiya waddiya chi-na-key
inna-banana-ga-watchio

[other verses I've forgotten]

The Chinaman, he up and died
and in his coffin he did lie
they shipped him off to old Japan
and that was the end of the Chinaman