Abby:

As you know, All Around my Hat as recorded by Steeleye Span was a collation; the tune and first verse came from AAMH, but the rest of the song was lifted from an unrelated piece, Farewell He; probably from versions collected in Hampshire and Dorset in the early years of the 20th century by Dr. Gardiner and the Hammond brothers (I expect I could find chapter and verse if needed).

The other set in the DT, All Around my Hat (I will wear the Green Willow), although it says "recorded by Steeleye Span", is completely different. No source is acknowledged, but it's actually the set recorded by Peter Kennedy from Harry Westaway at Belstone, Devon, in 1951, and published in Kennedy's Folksongs of Britain and Ireland. I'm afraid that I'm too young to know what was being sung round Cambridge in the late '50s!

The parody of the early 19th century can be seen at the Bodleian:

All around my hat I wear a green willow Printed by E. Keys, 7, James-street, Devonport. Firth b.27(536)

All around my hat I'll wear the green willow Printed between 1819 and 1844 by J. Pitts, 6, Great St. Andrew Street, Seven Dials [London]. Harding B 11(38)

If you remove the "cockney" patter and the vegetables, you wind up with something very close to Harry Westaway's set. Whether his descends from the stage song (with the extraneous material removed) or from its (putative) immediate predecessor, I can't say at the moment. I imagine that it could go either way.

I don't think that Chappell quoted a text at all, though as I said I haven't seen his earlier book. Kennedy's reference to The Budgen is a Delicate Trade is, I'm convinced, a mistake. The tune Chappell quoted as "the original of All Around My Hat" (as reprinted in JFSS) is indeed essentially the same as the tune we are familiar with.

I didn't have access to the Roud Index last time this thread came around. I see, though, that he classes The Nobleman's Wedding and All Around My Hat under the same heading (Roud 367, Laws P31); which makes sense. There would be a "missing link", presumably, between the former and the latter. Maybe that's something on the lines of the Westaway set; or maybe his was a descendant of the latter with the veggies and the chat dropped (not unprecedented, after all); I really don't know.