Got 10 hits on the Bodleian website, Catnach, Fortey, Pitts, Walker and some without imprint.

Yes there are tunes. There are a few trad versions. There's a Bedfordshire version in Hamer's Garner's Gay, p48 with tune. There's what appears to be the original sheet music in Speaight's 'Bawdy Songs of the Early Music Hall', along with a bawdy parody called 'The Drummer's Stick'. The original is written and sung by Mr Thomas Hudson 1791-1844 at London Concerts. There's no date on it but I'd guess about 1840. It's Roud 1076 by the way.

Hudson died in 1844 but this does not correspond with the publishers's dates. Henry White at 337 Oxford Street is the publisher and my dates for White at this number are 1849-57. White was at 350 Oxford St from 1838-49 but these dates are only approximate and he could have had both outlets running at once.