Les B- I wonder how many hits one would get on the little parlor organs. They went under various names; that may complicate the search.

Gilmore's (Lambert) sheet music first published in March, 1863 (actual release in July?- Fuld). It credited Gilmore with both words and music. A printer named Daly released the song also about July, under the name "Johnny Fill Up the Bowl," a fast bit of thievery.
Fuld says there is no tune relationship with "Johnny, I Hardly Knew Ye," J. B. Geoghegan, deposited British Museum, 1867.

Jack, you are correct about errors in the Fennell essay.

Gilmore's Band enlisted 16 Sept. 1861, attached to 24th Mass. Infantry. The musicians also acted as stretcher-bearers in battles at Bull Run, Antietam, Fredericksburg, Gettysburg, etc.
In August 1862, Congress discharged most military bands to lower costs. The governor of Massachusetts, however, charged Gilmore with re-organizing the State's military bands. In March, 1864, Gilmore was sent to New Orleans to oversee music for the inauguration of Louisiana's new governor. "This event, with its chorus of 6000, band of 500, cannons and anvils, foreshadowed Gilmore's later monster concerts."
The above from Univ. Maryland Archives summary. No mention of compositions.

Gilmore, if articles are to be believed, gave several answers to questioners when asked about his tune source, seemingly some facetiously.