Precisely the first part of "The Wearing of the Green" is used for the song "Benny Havens Oh!" composed and sung at West Point by Arnold O'Brien before 1842.

https://babel.hathitrust.org/cgi/pt?id=mdp.39015096411270&view=1up&seq=6

The entire tune seems to have been published (quite recognizably) as "To Ballance a Straw" in 1747.

The tune, the "Wearing of the Green" title, and the anonymous words of the broadside posted by Gargoyle appeared in The Citizen; or Dublin Monthly Magazine in Jan., 1841 (Vol. III, p. 65). The tune, nearly identical to the current version, has been slightly modified by the editor on the basis of various oral-traditional versions;

"We cannot trace, from remembrance, the ballad which was sung with it. Scraps here and there we can recollect....Some make 'Buonoparte' [sic] the hero of the song - but 'tis oftener, and we think more truly 'Napper Tandy' - for he was an Irishman - and although he has not left behind him a pure reputation for patriotism, yet, doubtless, he was once, in his day, admired and trusted by the people. Here is one version of four of the lines:

      I met with Buonoparte, he took me by the hand,
      Saying 'how is old Ireland and how does she stand,'
      "'Tis the most distressed country that ever I have seen,
      They are hanging men and WOMEN for the wearing of the green."

"Others will have them thus:--

      I met with Napper Tandy, he took me by the hand,
      Saying, 'how is old Ireland, my own dear native land?'
      "'Tis the most distressed country that ever yet was seen,
      They are hanging men and WOMEN there for wearing of the green."'


The anonymous editor (who takes credit for the broadside lyrics) associates the air with a song sung by Sarah Curran (1782-1808), fiancee of Robert Emmet, which he identifies as "Green on the Cape." (He appears to have heard her himself.)

Those lyrics are here, from an American broadside of 1829-34:

https://www.americanantiquarian.org/thomasballads/items/show/51