The song was popular in the 1930s as well as the 1950s, sung in primary school and in scout gatherings, especially the parodies. Its continuing popularity needs not have been bolstered by any particular recording since everyone already knew it. Sinsul, your son's parody was one of those we sang; I remember the green oysters. My grandfather sang it; so it was in the western repetoire at the end of the 19th C. and parodies were sung by my father who was in WW1.
None of this helps with the origin, however.