Well, Bob Davenport certainly didn't get the song from the Black Family! Their original note, not quoted on that website, was as follows:

"A once popular music hall ditty, now associated with Bob Davenport, but with verses collected by John Howson in Haughley, Suffolk. Learned from the singing of Dave MacLurg from Magilligan, Co. Derry."

It isn't entirely clear whether they mean that their set (via Dave MacLurg), or Bob's, derived in whole or part from the unidentified Suffolk one.

The full text and tune of Percy Webb's (Blaxhall, Suffolk) set appears in Ginette Dunn, The Fellowship of Song (London: Croom Helm, 1980), pp 70-71. It's rather different from the form Bob sang, but both follow the same pattern. Walter Pardon also had a fragmentary version. Peter Kennedy lists a few further examples, from Kent, Norfolk, Devon and Shropshire.

Some years ago, a further text was posted here, taken from a Dublin City Ramblers record; but with no indication as to where they got it. Quite possibly from the Black record. See thread Wheel the Perambulator John (with some rather strange typos).