'Follow Me up to Carlow' is a fairly modern song in any case, though set to an older (Scottish) tune. See thread  Origins: Follow me up to Carlow for details.

'Liza' and 'Eliza' were very common abbreviations of 'Elizabeth' during the 19th and early 20th centuries, and are becoming quite popular again nowadays. MacDonagh prints

I'll have Moonshine for Liza, and Moonshine for May,
Moonshine for Lu and she'll sing all the day;
Moonshine for my breakfast, moonshine for my tea,
Moonshine my hearties, it's Moonshine for me.

I can't think of any example of 'Lou' or 'Lu' in traditional song outside the USA / Canada, for what that's worth.

Galvin's "me" is just the way he pronounced "my", of course. "Tay" is the old pronounciation of "tea"; though that is still widely used in Ireland, it spoils the rhyme here and obviously wasn't intended by whoever wrote the lines. MacDonagh has a couple of rather mannered spellings: "mannie" for "many", and "counterie" for "country"; these may derive from his own fancy.

So far I see no reason to think that this isn't an American song; albeit containing some lines from earlier British and/or Anglo-Irish material. When I have time, I ought to post the tune as published by MacDonagh (and presumably provided, though no details are given, by Darley) to see if it is the same as the American examples of the 1920s.