Both references to the song from Fresno State imply they only contain the words and no tune (link); same with the VWML (link)

Purely assumption, but it seems to work fairly well in what I think of as a standard ballad timing. Each line has 14 or 16 (ish) beats; for the most part, each syllable gets one beat. Then, draw out the line as needed to match that (usually in the middle or at the end). I've written out two versions, one with 14 beats and one with 16; after doing both, I like the latter more, but I figured I'd leave both.

My main problem has been that the last line hasn't quite broken in a way that feels natural to me fully pronouncing every word, so I chopped a syllable off in each version (once in Michael and once in Jehovah, though I suppose Fagan could also be contracted; I just felt shortening Michael led the song to a more consistent pattern). You'll notice the main difference was just extending the last word, so if there's elements of one you like more than the other, just combine them.

14 beats per line

  Thrice has the English hangman sailed thro' Dublin Bay
  --1-----2---3---4--5---6-7-8-9---10----11----12-13-14-
  Each visit's been to take the life of some doom'd man away
  -1----2-3-----4---5---6---7----8---9---10----11---12--13-14
  Joe Brady and Dan Curley they have journeyed on before
  -1--2-3----4---5---6--7---8----9----10---11--12--13-14
  Michael Fagan has today gone forth to seek Je'ovah's shore
  --1--2---3-4---5---6-7---8----9----10--11---12-13-----14--

16 beats per line

  Thrice has the English hangman sailed thro' Dublin Ba-a-ay-
  --1-----2---3---4--5---6-7-8-9---10----11----12-13-14-15-16
  Each visit's been to take the life of some doom'd man aw-a--a-ay-
  -1----2-3-----4---5---6---7----8---9---10--11---12----13-14-15-16
  Joe Brady and Dan Cu-ur-ley they have journeyed on befo-o-re
  -1--2-3----4---5---6--7--8----9----10---11--12--13-14-15-16-
  Mich'l Fagan has today gone forth to seek Jehovah's sho-o-re
  --1-----2-3---4--5-6----7-----8---9---10--11-12-13--14-15-16

Again, this is total guess-work. Someone out there in Mudcat-land may know the tune themselves or find a text, but I didn't see any known modern recordings on Discogs or YouTube, or see anything that immediately jumped at me as having a tune included.