The song is printed in Campbell's collected poems (which are all in a similar bombastic style - even though I could once have bought a copy for 3 quid, I passed). It's easy enough to date when he wrote it - he was in Germany at the time. He had never been to Ireland at that point (not sure if he ever did) - born in the west of Scotland and educated in Edinburgh.

Thomas Campbell on Wikipedia

The tune is "The Banks of the Devon", a mega-hit of the time thanks to Burns's words for it. Burns got it on his one and only visit to the Highlands (I think it's the only tune he collected there) and he found it used for a Gaelic Jacobite song on the '45, which unusually for such songs was written fairly near the time of the events it describes. So the tune is Scottish and must have been around in the middle of the 18th century.

"Banks of the Devon" was used for a lot of broadsides. One much better than "The Bard of Armagh", though very little known, is a song that sympathizes with the French prisoners of the Napoleonic War interned near Edinburgh. I think John Leyden wrote it, but the author stayed anonymous.

Esk Mill