There is a book Four Irish Songs - By C. Milligan Fox - Words by Edith Wheeler and Alice Milligan. The Connacht Caoine (in Irish) by Tadhg O' Donnchadha. Illustrated by Seaghan MacCathmhaiol (Dublin, 1907) for sale on Abebooks and entry in the Royal Dublin Society library that contains the song and seems to confirm the Edith Wheeler origin for the song.

Since the sale copy is £115 plus shipping and Dublin is far away I won't be checking this, but maybe someone in Dublin could confirm it.

Mick