On a web page of John McCormack's recordings, I found other references to "Milligan":

Foggy Dew, The ("A' down the hill I went one morn...")(E. Milligan/C. Milligan Fox, arr. Spencer Clay)B 12767-1 (3 January 1913)

By the Shortcut to the Rosses (Nora Hopper/Old Irish Air, arr. C. Milligan Fox) BVE 41546-1, -2 (13 January 1928)

So there is an E. Milligan and a C. Milligan Fox...

I also found this reference on a web page for poems by Ethna Carbery (Anna MacManus), Click Here who died in 1902 : "Mrs. C. MILLIGAN FOX is arranging to have the greater number of the poems in book fitted with airs from Petrie's collection."
You might be interested also in seeing the lyrics of Rody M'Corely, as written by Ethna Carbery Click Here.

Alice