Hi Joe,
I can take it back to 1832 as it's listed in Catnach's catalogue of that year. However at the BL there is 'The Green Linnet' printed at Belfast in 1828. I haven't got a copy yet but its ref is BL 1078 k.5.40.

There are other 'Green Linnets' but these are later (one printed by Glasgow Poet's Box in 1852, and another by P. Hanley in 1883) however I think it very likely that the Belfast printing is our song.

Zimmerman c1830, broadside by W Kelly of Waterford you already have.

Keep safe
Steve