There are 67 entries in the Roud Broadside Index for this song. Most are from England, but about 10 are Scottish and 13 American. None are Irish, so I don't see the basis for saying it's probably Irish.

The earliest dated printing is from Scotland, a chapbook printed by J. Fraser of Stirling in 1817: https://digital.nls.uk/chapbooks-printed-in-scotland/archive/104186909.

Another Scottish printing is the one that attributes it to James Watt of Paisley: http://ballads.bodleian.ox.ac.uk/view/635ca286-d3a1-4d07-ac64-90d9fa72f626/. Whilst obviously one shouldn't necessarily take attributions like this at their face value, this one reads plausibly to me and the Poet's Box in Glasgow was near enough to Paisley(!).

However, the best summary of information about this song in in Pete Wood's "The green linnet: Napoloeonic songs from the French Wars to the present day" 2015.