All of this could go back to "The Nighingale's Song," or "The Souldiers rare Musick and Maides recreation," from 1682-1684, posted by Bruce Olsen in thread 3646: Nightingale . Now whether the tune for "One Morning in May" is the same- don't know. See the thread for more reliable information from Bruce O.
The song's descendants have been collected many times in America and all parts of the British Isles under the title "One Morning in May" or similar. I can't find a version in Randolph or Brown with "Merry Month of May" as title or first line; I would guess that Dominic Behan and others are referring to "One Morning in May"
The thread cited above (3646) has much information on the songs. I believe Bruce O says "The Bold Grenadier" is another song, but "a brave Grenadier" does appear, rather than a soldier, volunteer, etc. in some versions of The Nightingale or One Morning.

I think you can mention almost anyone who has ever done folk or bluegrass as having sung this song in one version or another. A Monroe version has been mentioned in one of the threads.