How to distinguish these two parts
Solution 1:
There would be no point mentioning the nursery rhymes if they were not also the subject of the sentence, but it was only the Grimm fairy stories which were translated (in 1823) – James Orchard's The Nursery Rhymes of England was published in 1842.
So the sentence can be broken down as
Both
(1) nursery rhymes, selected by James Orchard for a folklore society in 1842,
and
(2) [a] collection of fairy stories which were written by Grimm Brothers, swiftly translated into English in 1823,
soon rocket to popularity with the young, quickly leading to new editions.