This was published November 1915, in the U.K. I believe:
(thanks books.google.com)

The late Edwin Greene, composer of that tremendously popular ballad "Sing me to Sleep," was one of those fortunate composers that have been justly rewarded for their work. Mr. Greene was in the hands of honest publishers, Phillips and Page, London, and received his regular royalty on every copy sold throughout the world. In fact, this one song supported him. Mr. Greene was an invalid nearly all his life, and the royalties brought him the comfort that he so much needed.

Another source at the same website says, He was self-taught.

And this from an Auckland, New Zealand obituary, shortly after his 1915 demise:
....During the past 25 years Mr. Greene had been a confirmed invalid, as the result of very severe operations, and he was ever subject to acute recurrences of illness, but with the help of his wonderful vitality he made a courageous struggle against physical misfortunes. It was out of this suffering that his musical renown directly sprang....By assiduous self-culture he qualified for the post of organist .... but his health soon broke down. His gift for song-writing was evinced early in life, although it was not until his affliction came that his special gift was developed on lines that met with immediate public recognition. It was while lying in a hospital ward that he wrote the words and music of "Springtide," in after years the composer's favorite....A beautiful setting of Tom Hood's pathetic verses, "I Remember," was next published, and was quickly followed by "Sing Me to Sleep," ...it leapt into popular favour at once. The whole English-speaking world was soon singing it, and other nations caught the infection, for the German, Hungarian, French, Italian, and other translations sold out rapidly. The publication assured Mr. Greene a regular income, his publisher paying him a handsome retaining fee so long as he reserved his compositions exclusively for the firm. No subsequent lyric from his pen achieved such fame, although all sold well."

The silence around his affliction is typical of the era, I suppose. Consumption?