"Tweedside" is on my website, as one of the tunes from Joseph Mitchell's "The Highland Fair". I can only vaguely remember "For Ireland I Won't Tell Her Name", but maybe. Was it one of those mid-19th-century Irish nationalist songs where borrowing English and Scottish tunes wholesale was the usual procedure?

The oldest known lament in Gaelic tradition (probably predating the divide into Irish and Scottish Gaelic) is probably "Druimin dubh", but nobody sings it any more either side of the Irish Sea.