Them fleeing to the colonies would be interesting. 1640s Virginia is not like 1770s America. The Dominion had only existed for a generation, for a start, and living conditions weren't great. If for some reason Charles and his family went there, they'd fade to insignificance rapidly - there wasn't enough of a society for them to live "the high life", there's a good chance several family members could easily die of disease, and indeed the colony could collapse trying to support them unless they were willing to live the life of nothing more than a penniless country gent. On top of that, if the King fled to the colonies it would likely see the first time that the English ever sent a military force to occupy their own colony, and if that happened the King frankly wouldn't have half the chance of escaping that he would anywhere in Europe - there'd just be nowhere for him to go and no-one to shelter him.
Now, Europe is different. Of course, Europe also has a million and one different options. You'd really need to specify when exactly you're proposing he flees to get an actual specific response, however...