The illustrator Charles Keeping included it in a wonderful collection of music hall and parlour songs, "Cockney Ding Dong," published by Kestrel Books (in association with EMI Publishing) in 1975. It's a well researched book, and the fact that the credit for this song is "collected by Charles Keeping" suggests to me that finding out who wrote it will not be easy.

My guess though is that it was indeed written. It's surely too polished a song to have just grown.