While I've been off checking references, most of the material I had intended to point to has been covered.  It remains only to mention a few small points:

The use of Irish names, locale and words such as shillelah (however spelt) are not in themselves indicators of Irish origins in a song.  "Stage Irish" songs, largely written in London, were popular from at least the 18th. century.  Some used existing melodies, others had new tunes made in an "Irish style".  By the 19th. century, the USA was also a major source of "Stage Irish" songs.

The earliest known examples of this song appear to have been found in England.  That doesn't necessarily mean that it didn't originate in Ireland, just that we don't have evidence going back that far as yet.

Malcolm