Rumor has it that when they didn't feel like singing "Galloping" they sang "Ballocky."   One of the worst things you could sing out in, say, 1922.

Wish I could add more beyond my feeling that "don't suck your thumbs" is probably euphemized from "pull out your thumbs"
(At least "Pull your thumb out" is a recognized cliche'.)

Hugill says that only "two or three" of his verses "have had to be altered slightly," so he wasn't obviously thinking of Randolph's version.