A fun way would be to have large numbers of Roundheads (including possibly Cromwell) escape to exile in New England instead of being executed or quietly reabsorbed into English society. New England very, very strongly supported the Lord Protector - in fact, New Haven ended up taking in several prominent regicides, and droves of young New Englanders enrolled in the New Model Army. Such a New England would probably conquer the New Netherlands sooner rather than later, after which they'll likely expand into much of OTL New Jersey and maybe into Pennsylvania until they hit Virginia and Maryland. They could well be rabidly Puritan (though absorbing the New Netherlands could hurt that, it wouldn't be impossible for the local Dutch to adopt Puritanism - there were strong Puritan-esque movements within the Dutch Reformed Church anyway; maybe that alone would be enough). Such a country would also be well-placed to absorb Huguenots.
Of course, they'll run into conflicts with Maryland and Virginia (I'm not sure they'd be able to stop themselves), and very likely New France. I don't know if they'll win more than they'll lose.