A version ("The cuckoo is a merry bird") in Opie's Oxford Dictionary of Nursery Rhymes (new ed., no. 121) has this verse:

The cukcoo comes in April,
She sings a song in May,
In June she beats upon the drum,
And then she'll fly away.

("The cuckoo is said to 'beat the drum' when he often falters and cries 'Cuck-cuck-cuck,' without the final syllable. This is said to be a sign of his impending departure."--Williams)

This (from Alfred Williams, Folk Songs of the Upper Thames, 1923, p. 165; no music) is a related song to CUCKOO'S A BONNY BIRD and CUCKOO SHE'S A PRETTY BIRD.

~Masato