The title, by the way actually though not in literal translation means "I wouldn't tell you her name if you gave me the whole of Ireland".
PW Joyce's Old Irish Music and Songs - here on the Internet Archive, to which you might throw in a tenner's worth of support if you can afford it - has Nancy the Pride of the West is; Joyce describes it as having the same air as "a beautiful song in Irish, to which I published… in Irish Music and Song.
Unfortunately, the same Irish Music and Song doesn't seem to have been added to that archive by anyone.