That's quite a tricky question, G!
In a sense the Royalists *did* win, in that the monarchy was restored shortly after the death of Cromwell.
It might be argued that had the Commonwealth continued in its original form (i.e., not so much a dictatorship by Cromwell), there would have been less cause for the English-speaking colonists in N. America to seek separation from the mother country over a hundred years later (a) because the religious sensibilities of the two "countries" would probably have had more in common and (b) the political views of the two would have been better attuned.
But hypothetical history is so difficult, and I'd love to know what others thinK!
PS. There's an interesting discussion of your question at
http://tinyurl.com/2sugs