It's perfectly possible that the song is a mid-nineteenth century stage piece; a good few "traditional" songs are exactly that, and nowadays it's increasingly easy to find background material that wasn't necessarily available to the folksong collectors of a century ago.
I know of only one tune found in tradition for this song, and it doesn't vary all that much. Beside the set that appeared in Heywood Sumner's The Besom Maker (1888; no source is named) and the modified form of it re-printed in Lucy Broadwood and J.A. Fuller Maitland's English County Songs (1893), there is one in Frank Kidson's Garland of English Folk Songs (1926) -which I haven't seen- and one in Colm O Lochlainn's Irish Street Songs (vol. 2, 1965), which came from Alice Deady of Waterford. This last was the source of Steeleye Span's arrangement of the song; again, the usual tune, but they interfered with it a bit to give it a faintly exotic rhythm of the sort that Bert Lloyd used to favour.
It's a great pity that images are not available at Levy; assuming for a moment that it's the same tune, I'd be particularly interested to see if the A sharp that Fuller Maitland (not Lucy Broadwood) "corrected" to A natural appears in the sheet music; Alice Deady's set, which was much more recent than the others, has (if transposed) the natural, which is what one might expect in a traditional song. It might be, however, that the sharp would be more characteristic of a composed piece; Fuller Maitland's alteration clearly reflected traditional practice, but may perhaps have been premature. Of course, it is perfectly possible that Alice Deady may have learned the song from the book, either directly or at one or more removes, so without more information we can't make any useful deductions on that score.
W.J. Florence was an actor, best-known in America, though I don't know if he was born there or not, and apparantly also a prominent Freemason. His wife was Malvina Pray, also a professional performer; her father, Samuel, had worked at the Broadway Theatre until his death in an accident involving stage-machinery (specifically, a curtain-roller).