My mother (born in 1919) taught me the first verse. Three others I got from ... I forget, but possibly Oscar Brand's Bawdy Songs. I just read a piece on the New Yorker website about gendered bathrooms, and thought of the verse that I sometimes quote, especially when liberating a men's room, which I have done a few times:

If the women's room is taken
Do not feel at all forsaken,
Never show a sign of sad defeat,
Try the men's room 'cross the hall
And if a man should have the call
He'll courteously relinquish you his seat.

As for the tune, Dvorak's "Humoresque," a century ago everyone knew it, and anyone who played the piano could play it.