Other oral versions and broadside copies are listed in the Roud Folk Song Index under number 1088. Tunes are not consistent, true; indeed, one of Percy Grainger's singers (Edgar Hyldon) actually knew two completely different ones, though apparently none of the words.

The song appeared on broadsides of the early and mid-19th century as 'The Jolly Waggoner' and 'The Warbling Waggoner'; although extant copies of the former title seem largely to be earlier, they include Bedford's additional verses, whereas editions of the latter do not. Various editions can be seen at  Bodleian Library Broadside Ballads:

The Warbling/Jolly Waggoner

None names a tune. Frank Kidson thought it probably a product of the late 18th century, and that seems likely enough; I'd suspect a stage origin, as is the case with so many songs of this kind, but that's only a guess. Mayhew mentioned it in London Labour and the London Poor, and W S Gilbert specified 'The Warbling Waggoner' as the tune for a song, 'When first I went a-governing', in his burlesque Ruy Blas: A Preposterous Piece of Nonsense for Private Representation (Warne's Christmas Annual, 1866) so presumably it was still well-known at that time. To what tune, though, I can't at the moment say.