My Mom taught it to us as:

There was an old man from Chinatown
His name was chittereye chee chi cho
His legs were long
His feet were short
This old man could laugh nor snort

Chittereye chee chi cho chigaloro
Bungo loro piggy wiggy waggo
Hoco poco hit 'im on the coco
Chittereye chittereye chee chi cho

China man had a wife
Led him such an aweful life
Cut his pigtails off so long
Sold them for a Chinese song

Chittereye chee chi cho chigaloro
Bungo loro piggy wiggy waggo
Hoco poco hit 'im on the coco
Chittereye chittereye chee chi cho

China man he did die
In his coffin he did lie
Shipped his body off to Japan
And that was the end of the China man

(chorus sung as fast as humanly possible)
Chittereye chee chi cho chigaloro
Bungo loro piggy wiggy waggo
Hoco poco hit 'im on the coco
Chittereye chittereye chee chi cho