It definitely goes back to 1944 Joe. Irwin Dash Music Company got copyright for an arrangement of it that year. But that's about as far back as it seems to go. I guess you could catch a lot of people out with that in a pub quiz.

I've never heard the original recording, or the Danny Kaye version, but I assume it is put across in the grand old tradition of the Edwardian music hall knees-up (a la Any Old Iron, Knees up Mother Brown, Don't Dilly Dally, etc which proliferated around 1908-12. Or maybe it's done as a parody, in Dick van Dyke-type Cockney?

Charles Keeping included it (erroneously?) in a collection published as "Cockney Ding Dong" (1975), where it was correctly credited it to Fred Heatherton.