Guest Student, you have picked a tough one.
See posts by Bruce O(lson) from 1999, above. He was an expert song researcher.
So all we know is that it appeared in a book in 1847, and later a remark was made that it was in an old broadside from the 18th c (disputed by Olson).

The comments about an old French carol may be wrong. Musical settings today are 19th c although one or more have borrowed French melodies (see posts above).

See comments by Malcolm Douglas (another careful authority) in the other thread- 42010. He found a couple (and gave a link) in the Bodleian Collection that are early 19th c. Thread 42010: Holly and the Ivy

All the comments in various websites about a pagan origin are nonsense.
There are other 'holly' songs, mentioned in a footnote to the carol in "The Oxford Book of Carols," 1929, p. 81.

Some 19th century sheet music can be found here (scroll down below the song text): Holly and the Ivy