The English Carol "Jacob's Ladder," in the "Oxford Book of Carols," bears the note "This is apparently a carol to which new words were fitted under the influence of the Methodist revival. It is printed here with its traditional melody, which Stainer made familiar in 1871."
The melody probably has been used for several songs and the traditional melody may be quite old.

"The Woodpecker Song" may be found at www.immortalia.com. and in the Mudcat DT; that version is probably 19th c., but I doubt that this is the one you mention.