Peter Kennedy and Seán O Baoill collected the song from Hudaí Ó Duibheanaigh in Rann na Feirste (aka Ranafast), Co Donegal in 1953. But Kennedy was not the first person to publish the lyrics. I'm looking for older sources and here is a score, with lyrics, from Carl Hardebeck. No publication date given; Hardebeck died in 1945.

Turlough O'Carolan composed a song to Brighid Ní Mhaille, but I haven't found out yet whether the tune and/or the words were the same as Bríd Óg Ní Mhaille. O'Carolan the harpist was a praise singer, so the lines about Bríd marrying someone else don't strike me as typical O'Carolan.