Not much mention of the tune except for what GUEST,thurg has to offer - (The Galway Shawl)
It might be of interest, for the record, that the Australian song "Moreton Bay" has been collected here with the same tune. Singer Alan Musgrove notes that "the tune is English". It was collected by Norm O'Connor from Simon McDonald of Creswick, Victoria in 1960. Simon McDonald learned it from his uncle Jack McDonald who was born in 1850.
Cheers, Joy