Karl Dallas gives it as "My darling sleeps in England" - the note in the book gives no further details about who got it from Mary Reynolds' singing, or where she'd have got it from.

Karl Dallas wrote in that note in the book: "Some have called the melody trite and 'Irish pub tenor' but these words and simple tune are strangely moving, especally as Mary sings them in her high sweet voice." It'd very likely have been the same Mary Reynolds who sings "Brennan on the Moor" on Topic's "The Folk Songs of Britain, Vol. 7. Fair Game and Foul" (which I've never heard), since Karl gives a specific mention of Topic Records in his Acknowledgements section.

Maybe you could ask Karl - I haven't got an address for him, but it shouldn't be hard to track him down.
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But I just did a Google for "My Darling Sleeps in England", and got this: MY DARLING SLEEPS IN ENGLAND - Irishman concerned about his girl in Birmingham during the bombing in the Second World War (known as "The Blitz") -- Mary REYNOLDS rec by Seamus Ennis, Mohill, Co Leitrim 1954: RPL 22028 (From here

Mind from that (it sounds as if whoever wrote it hadn't actually heard the song ("Irishman concerned about his girl in Birmingham"), or maybe Mary had two versions.)

Frank Harte would be the man to ask though.