Stanley Holloway certainly recorded it, since that's where I learned it (all the verses except the MP one were on it). My version was on a cassette, but my guess is that it was copied from a 78.

Many others were there, too: Down at the Old Bull and Bush; I was Walking through the Park, one day; Any Old Iron? She's my Lady Love,
etc.

If any one can locate that 78, I'd be very grateful.