to be even pickier, the tune seems to have been written in 1534. So since Elizabeth didn't become queen till 1558, it wasn't Elizabethan either way.
Youtube does have it posted as by the Kings singers, so it may well be so.
http://www.youtube.com/watch?v=w7tFM...eature=related
The nice little wiki about him says that song is intended to copy the clucking of chickens, which is quite sweet.
http://en.wikipedia.org/wiki/Pierre_Passereau