This ballad dates back to at least the time of Charles II. It had several titles including "Bonny, bonny Bird" and is printed in Chappell's Popular Music of the Olden Time.
This version comes from Sharp's One Hundred English Folksongs. He made "one or two slight alterations" to the words to suit editing tastes of 1916.
I knew the tune because it was used by Ralph Vaughan Williams, along with Green Bushes in the Intermezzo movement of The English Folksong Suite.