Okay according to my indexes the tune is available in various volumes.

Ed Cray in The Erotic Muse gives an American version to the ubiquitous 'Vilikins' tune which is related to the American bawdy song 'Charlotte the Harlot'. He states though 'Originally from Great Britain' However the tune of the 'all in blue, all in blue, I said 'How do you do' bit is the same tune as I sing.

One of the earliest versions I have is in Nettleingham's 'Tommy's Tunes' from WWI. Unfortunately the way the tune is set out in harmonies (piano music) I haven't the skill to work out what the tune is. Above the score it states 'TUNE: Special' which I take to mean it has its own tune. It could be related to Vilikins but that's just a guess. It has 11 verses and a sort of epilogue that has the designated tune of 'Farmer Giles' which I do believe is the Vilikins tune.