I don't know much about the United Irishmen, but I doubt that there is any connection to this particular song beyond the name; there were other Captain Thunderbolts, too.  In Ireland, there was John Doherty, a highwayman active between c.1810 and 1818.  After a fairly high-profile career as a robber, during which he acquired the kind of romantic, "Robin Hood" reputation often attached to such people, he fled to America; after some moving around he assumed the name of John Wilson, and quietly practiced medicine (without a license) in Vermont until his death in 1847.  Slightly later, the Australian bushranger Frederic Ward (1836-1870) adopted the same nom de guerre.

It is not impossible that The Shannon Side may refer to Doherty; I don't know when it first appeared, but most of the broadside examples at  Bodleian Library Broadsides  are of the first half of the 19th century.  Richard C. Simmons'  Eighteenth Century Cheap Print: A Finding Aid,  however, gives a tentative location and date of Dublin, 1780 for the first printing (second example shown below), so it may be more likely that Doherty's pseudonym came from the song.  The occasional verse from it has at times crept into sets of Down By a Riverside/ Abroad As I Was Walking in English tradition; James Reeves (The Idiom of the People, 1958) said, of Captain Thunderbolt, "supposedly the devil", but I don't know how much credence I'd attach to that.  Mind you, Captain Thunderbolt appears as Saint George's opponent (usually the Turkish Knight or similar) in the  Ilmington, Warwickshire Mummers' Play  (version noted c.1913).

The broadside examples don't vary a great deal, textually; here are two:

Shannon side  Printed between 1819 and 1844 by J. Pitts, Wholesale Toy and Marble Warehouse, 6, Great St. Andrews Street, Seven Dials, London.
A new song, called Shannon side  Printer and date unknown (but see above).

Can't help with a set starting with that verse, or referring to the Roe, I'm afraid.

Ezio's transcription isn't marked as harvested; just in case, I'll indicate corrections to a few minor typos and omissions:

Verse 1, lines 1 and 2 (missing above, here taken from a broadside copy):

It was in the month of April, one morning by the dawn,
When violets and cowslips were strewed on the lawn,

v3 line 2:  young man

v4 line 3 and 4:

Now you have had your will of me
And do not leave me...

v5 line 1:  easy

v8 line 4:  wealthy grazier's...