That's odd, Jon! Rollins gives the following entries for the earlier period.
Miller and King (1625)
The Miller and the Kinge (1625, probably same piece)
A Miller I am (1564)
The Miller's Daughter of Mannchester (1581).

It's well worth reading Child's headnotes to throw some light on origins. Generally Scandinavian versions are fairly recent and from oral tradition (Norse, Danish, Swedish) but he refers to one Icelandic version of the 17th century. That still leaves the very real possibility of an English origin and Smith as the author. I fully believe 17thc broadside ballads like Cruel Mother, Demon Lover, Lord Thomas & Fair Eleanor are the originals, so why not this one?

Rimbault may well have had a broadside of the same date 1656 printed by Grove in his possession. These items are extremely valuable and I'm certain there are still many out there in private collections . Where does it state that the broadside was from the Antony Wood Collection. Apart from anything else Wood may have sold/swapped his copy before it came to the Bodl. I cannot conceive that a highly esteemed antiquarian would lie about such a thing. Why not just say he got his copy from one of the other contemporary sources?