A confusing aspect of 17thc pieces is those items that appeared in anthologies were often claimed by others to have been written by the compilers, as is often the case with D'Urfey's Pills. It may well be that Smith was simply one of the compilers. Child opines that the ballad may already have existed and that Smith might have added the 3 'burlesque' stanzas. Whilst this is possible it is still only conjecture.

BTW I have Grove's dates as 1620-1655. It might be possible to find out when he died, but often when a printer died his family carried on the press using the established name.

Going back to the origin question, where Scandinavian versions occur the vast majority have usually been translated into English in the 18th and 19th centuries, but it is quite likely that a few went in the opposite direction. In fact I'm pretty certain those few Danish versions of The Cruel Mother derive directly from Grundtvig's 1840 translation from Engelske into Danske.