Just for comparison, here are the songs in the Database:

CUCKOOS NEST - No source given. A story of a courtship and marriage ending with the line "And he often claps his hand on her cuckoo's nest."

CUCKOO'S NEST, recorded by Jean Redpath. It starts, "There is a thornbush in oor kailyard" and ends "I left her wi' the makings of a young cuckoo."

THE CUCKOO'S NEST by (John Shiels). This version plays with big words: "If experienced in ontology, relate without tautology/The pristine aestheology of my cuckoo's nest."

THE CUCKOO's NEST (2) begins "There's a pombush in the garden where the lads and lassies meet"

THE CUCKOO'S NEST (3), a bawdy cowboy version beginning, "Some like a girl that's pretty in the face"

THE MAGPIE 2, an Irish version from Peter Kennedy's book, "Folksongs of Britain and Ireland." With the refrain "I will l'ave you down to rest in the magpies's nest.

None of these seems to be the song Eclipse is looking for. I also searched the Forum for "Cuckoo's Nest," and everything I found there seems irrelevant. Also, using "cuckoo's nest" + "climb * tree" came up with nothing relevant from Google.