The British Library has the sheet music, as follows:

MICK McGILLIGAN'S BALL.
Written by Michael Casey, ©1948

The date of 1948 would mean it is NOT in the public domain—unless that was a reprint of a much older song—but I think the 1948 date is probably accurate.

WorldCat.org also shows that it's included in the following songbooks:

Irish Music Hall : An Extravaganza of Lovely Ballads and Humorous Ditties, Including the Songs of George M. Cohan, Thomas Moore, and Others.
(Melville, N.Y. : MCA/Mills, [197-?])

40 Irish Hits of Our Times.
New York : Leeds Music Corp., ©1956.