A separate song on the face of it; though perhaps the tune would tell us if there is a connection. The song is a round, and quite well-known in various forms. I recall it from Scout camps in the mid 1960s, for instance, as

From out the battered elm tree
The owl's cry we hear
And from the distant forest
The cuckoo answers clear:
Cuckoo, cuckoo, cuckoo_
Cuckoo, cuckoo, cuckoo_

In that case, the tune resembled a sequence from The Trout. Probably there are other tunes, and maybe it is a shortened form of Judith's song.

See also thread from the fading ditto sheets, where a two-verse example is quoted with the suggestion that it's a translation from French.