I learned it this way in the early 70's, it was definitely not to tune of Blow The Man Down. There was a girl at school who knew a lot of these types of rhymes. I can't remember her name at all. We were not best friends or anything like that.

That Silly Chinaman

My mother told me open the door, I did not, did not want to
I opened the door, he fell on the floor, that silly Chinaman
My mother told to take off his coat, I did not, did not want to
I took off this coat he smelled like a goat, that silly Chinaman
My mother told me to give him a bath, I did not, did not want to,
I gave him a bath, he showed me his ass, that silly Chinaman