The musical play "Chu Chin Chow," by Australian-born Oscar Asche (lyricist) and Frederic Norton (composer), opened in London in 1916 and ran for nearly 5 years. There were later productions in the US and Australia, but they were less successful. See PeoplePlay UK, Guide to the Musical Theatre, and National Library of Australia.

The Internet Movie Database (IMDb) lists 2 film versions made in 1923 (a silent musical?) and 1934.

I can't definitely identify any song that was the source of CHING CHONG CHINAMAN, but the most likely-sounding title is I AM CHU CHIN CHOW OF CHINA.