I can't help with the origin of this song.

But I would like to highlight that a beautiful version of Turtle Dove can also be found on "Olde English Madrigals & Folk Songs at Ely Cathedral," by John Rutter and the Cambridge Singers.

Rutter scores a very short version of the song (5 verses, no chorus), but it is gorgeously sung.

Also on the album are other madrigals by Greaves, Morley, Weekles, etc., as well as British traditionals such as "I Love My Love" and "Bushes and Briars." Rutter also put out two choral score books for all the songs on the album, which are very handy.