Summary:
1. The anonymous song sung by Disley is most likely a parody or derivative of an earlier song, as quoted in two books listed above.

2. The poem from the books is also anonymous. The first verse is the same, but the second seems to be the start of a totally different story.

3. Similar verses, e.g. Tottie, were written and published in the 1880's by George R. Sims (aka Dagonet). His best known work is Christmas Day in the Workhouse (the original, not the well known parody)

Therefore is is quite possible that he wrote the original. But no proof yet.