Helen wrote: On the Traditional Tune Archive a note on the song Sumer is icumen in states that the melody was originally an Irish tune, collected by Bunting.

In other words, Edward Bunting, The Ancient Music of Ireland, 1840.

I don't find "Sumer" in its contents list by that title, but that's not surprising, because Bunting pretends everything is Irish and names it accordingly. However, the attributions are revealing. The large majority of the airs are described as "Very ancient, author and date unknown" -- but were collected in 1792 or later. Bunting has no proof of anything; his phrase "very ancient..." merely means "source not known to the harper who played it." The next-most-significant source is O'Carolan. A few other sources are listed, but the overwhelming majority are one of those two.

Very few have words -- not even things that everyone knows are songs, e.g. "Banks of Claudy." Given that they have been rearranged for piano, it will be hard to identify any tune with the manuscript melody, which has a different number of parts. I'm still looking. But I don't believe in the strength of five hundred years of conjectures.