I, too, have tried to get the lyrics to this song. I learned it in grades 4 & 5 (1954-1955) in Brockton, MA - B.B. Russell School.
What I've found so far:
The tune was used as the prevailing theme song for John Wayne/Maureen O'Hara "The Quiet Man" (1952). That film calls it "The Prelude To The Big Fight", by Victor Young. Great movie. If you've never seen it, you should.
Somebody once told me it was called "The Meadow" or "The Lea", but I never had any luck with those words in titles.
Trivia I ran across: In searching several old children's song books, I also learned that many of the very old ditties and songs we learned as kids were written by a man named Englebert Humperdink.
I plan to continue looking, and if I get the answer, I'll let you know. Good luck in your search.
Send your email address if you wish further correspondence on this subject. I'm not looking for pen pals, or anything else.
Bill