Wikipedia says Cecil Sheridan wrote "Mick McGilligan's Daughter."

However, Trinity College Library Dublin has a copy of the sheet music, which is described thus in its catalogue:

Mick McGilligan's Daughter, Mary Anne
words and music by Joseph M. Crofts.
Dublin : Walton's Piano & Musical Instrument Galleries, [1958?]

Likewise, the British Library says:

Mick McGilligan's Daughter, Mary Anne.
Joseph Crofts
c1949.

Other songs I have found attributed to Joseph M. Crofts, in the years 1945-1952:

'Twas One of Those Dreams
Back to Dear Old Tipperary
Bould Thady Quill
Come to the Dance at Kilbecanty Fair
Could I But Hear
Dry Be That Tear
Eileen Macmanus
Evelyn
Evermore
Goodnight Molly Darling
Hills of Connemara
How Dear to Me the Hour
In the Sweet Lovely Vale of Adare
Love of My Heart
My Dark Slender Boy
My Darling Child
My Darling Mary
Noreen Ban
O, Molly, I Can't Say You're Honest
On the Banks of the Calm Winding Feale
Shamus O'Brien
The Carlow Maid
The Chimes of Cove
The Irish Peasant Girl
The Irish Rover
The Ploughman
The Wild Colonial Boy
Tipperary Far Away