There's nothing Irish or Irish American about this song; it is very firmly English, and belongs specifically to Lancashire. It is popularly known nowadays as The Four Loom Weaver, under which name Ewan MacColl recorded it from Becket Whitehead of Delph, Oldham, Lancashire, in 1947 (The Singing Island, 1960, 38). Mr Whitehead was nearly 80 years old, and had learned the song from his father, a handloom weaver and active Chartist.

You can see broadside editions of the song from 1850 at  Bodleian Library Broadside Ballads:

Jone o' Grinfield

There was a whole series of Greenfield or Grinfilt songs, following from James Butterworth's original Jone o' Grinfilt. Various aspects of the song have been discussed here in the past, particularly, I think, the nature of "Waterloo Porridge". The search engine will find all sorts of stuff if asked.

"Tarry" here has nothing at all to do with "Tarriers", which was discussed at some length here a few years ago: Help: What is a tarrier.