Apologies for misleading with the Sims reference earlier. Now I've had a chance to check Disher and Kilgarriff the latter seems convinced that Sims is not the author, although in order for him to make this statement, he must have seen similar references to what I vaguely remember.

Kilgarriff refers to the Billy Bennett (1887-1942) repertoire, but also gives it in Elsa Lanchester's repertoire (1902-1986) alongside such pieces as The Old Dun Cow and Your baby has gorn dahn the plughole. He dates it as c1901, presumably after Disher. It might have some relevance in dating the other 2 Cockney songs in Lanchester's repertoire. The fact that it doesn't appear on broadsides would favour a later rather than earlier date around that put forward by Kilgarriff. The c1900 equivalent of broadsides was booklets of 30-40 popular songs and it may well turn up in one of these.