Andrew Kuntz (Fiddlers Companion) quotes O'Neill (Waifs and Strays of Gaelic Melody,1922) as follows:

"Classed as a street ballad in Halliday Sparling's Irish Minstrelsy, London 1887, the editor adds, in a note on page 366, 'Johnny I Hardly Knew Ye! This favorite old song is here for the first time given complete. It dates from the beginning of the present century, when Irish regiments were so extensively raised for the East India service.' "

Whether Sparling had any evidence to support his claim, I wouldn't know; nor whether the tune he had in mind (if any; he didn't print any tunes) was the same as, or similar to, Gilmore's. At any rate, John Anderson My Jo seems convincing enough as the melodic ancestor to When Johnny Comes Marching Home; it's much too close for coincidence.

Simpson (The British Broadside Ballad and Its Music, 1966, 394-5) points out that that tune is "to be equated with I am the Duke of Norfolk" [or Paul's Steeple] "although its melodic line has undergone independent development". As has already been mentioned, William Chappell (Popular Music of the Olden Time, refs above), considered I am the Duke of Norfolk to be the older form; Alfred Moffat (Minstrelsy of Ireland, 3rd edn, nd, 136-7) is inclined to dismiss Chappell, but this turns solely on the date of the Skene MS, about which I think there is still some debate, and in which the tune 'Johnne Andersonne My Jo' first appears "in amateurishly notated tablature" (Simpson, p 394: the staff notation he prints is Dauney's 1838 'regularised' transcription).

The 'I am the Duke of Norfolk' lines quoted above don't invalidate the argument, as they probably don't belong to the original (lost) song: the verse is part of a Suffolk harvest tradition, and was originally quoted in The Suffolk Garland, 1818. Chappell made it fit the melody, but that doesn't prove anything in particular. The respective ages of the two tunes are moot, but there seems no reason to doubt their relationship. Chappell sometimes overdid things, but Simpson is pretty reliable. Cruiskeen Lawn, which is very similar to I am the Duke of Norfolk (and may well derive from it), doesn't appear in print until Coffey's Beggars Wedding (edition of 1731).

The SMM suggestion of a liturgical source derives from Percy, Reliques, Vol II part 2 number 2: he provided no specifics, and the proposition seems doubtful.

I'd be inclined to suspect Johnny I Hardly Knew Ye of being an ironic parody of When Johnny Comes Marching Home rather than the reverse, but the question of dates, as mentioned by Sparling, remains to be dealt with and I may be quite wrong on that score. At all events, there is no particular need for the tune to have been learned by Gilmore during his youth in Dublin, as O'Neill also speculated. That would be perfectly possible, of course; but John Anderson was well enough known in America at that time.