The "hey nonny"s I've encountered are in a round, which I learned in oral tradition, and which was published in 'Melismata' (1611).

He that will an Alehouse Keep must have three things in store:
a chamber and a featherbed, a chimney and a
hey nonny nonny, hey nonny nonny, hey nonny no, hey nonny no, hey nonny no.

The hocket or 'catch' is more subtle than some:
'H / ey' coincides with 'st / ORE' .