What Kilgarriff says:

"Your Baby 'as Gown Dahn the Plug-'Ole." (Spade) DAS/CCC. 1944.
Elsa Lanchester. From "Ditties from the Ditty Box." Jack Spade, a composite pseudonym of Elton Box, Desmond Cox and Lewis Ilda, the last-named being the publisher Irwin Dash. "It seems likely that this song is a burlesque with no authenic traditional origin. It was recorded by Elsa Lanchester in 1951. The U. S. version, "Your Baby Has Gone Down the Drainpipe," is by Morey Amsterdam: Kay Patrick (FEL/DFH, 1946).

Michael Kilgarriff, 1998, "Sing Us One of the Old Songs," A Guide to Popular Song 1860-1920. Oxford Univ. Press