The original tune is an older Irish melody which has various names and words set to it. And has been arranged or had lyrics written in the 20th century.

Joe Offer's website has some good info and I have also had a dig around.

"The melody is traditional, and was taken from the Stanford-Petrie Collection, where it appears as I rise in the morning with my heart full of woe: a Cavan air. Also known as The Coola Shore.

The tune given here is as published in The New National Songbook. The note-values have been lengthened and smoothened rather from the set in Petrie. A fairly close relative of the tune seems to have wound up in America attached to Pretty Saro." -- Joe Offer

My Love's An Arbutus:
Lyric by Alfred Perceval Graves.
Tune traditional (Cavan): I rise in the morning with my heart full of woe/ The Coola Shore.
The piano part is composed, but not the original melody, by Charles Villers Stanford (1852-1924).

From a sheet music photocopy:

The Fisherman's Night Song: Irish trad. with lyrics by L. A. G. Strong
This was first published, possibly, arranged by Herbert Horrocks in 1958.

A little info about him:
"L.A. Strong (1896-1958) [poet etc] was born in Plymouth, of a half-Irish father and Irish mother, and was educated at Brighton College (where in later life he was a governor) and at Wadham College, Oxford (Open Classical Scholar). There he came under the influence of W. B. Yeats."