Irish? Probably not. It seems to have been found most often in tradition in England and Scotland (rather more frequently in the former country, so far as I can tell); also in Canada, Ireland and the USA.

You can see a mid-19th century broadside edition from Harkness of Preston at  Bodleian Library Broadside Ballads:

Never maids wed an old man

Note the "hey down derry down" refrain in the broadside, likely enough the origin of "hey ding a doorum da".