Before Charles I is captured, he and a large number of Royalists flee to the colonies and set up a new government.
There's really absolutely no way that this would happen. For a start, the colonies were only about 50 years old at this point, and Charles and his "government" would be forced to live in squalor, unable to effectively govern further than a few nearby settlements. Furthermore, if he does this he loses all credibility he has, forever. To the European monarchs who might have sheltered him and supported any attempts he might have made at restoration, he's thrown away his pride and his royalty by cowering in America rather than relying on their hospitality. To his enemies, he's running away from the problem and is no longer a worthy adversary, rather he's someone who should be hunted down like a dog. To his supporters, he's no more than a desperate man who's given up.
On top of this, he faces two major problems by going to America. One is that he risks bringing the Civil War to the colonies. Not everyone in the colonies is going to be cowed into supporting him, and the southernmost colonies and perhaps some of the northern ones are likely to turn their backs on him in favour of supporting the "legitimate" government. Remember that OTL only Virginia declared support for him after he lost the war. Secondly, he's fleeing to a land filled with only about 60-70,000 people. He's never going to have the maritime resources to transport an army, and to be brutally honest, Cromwell would respond to this by laughing at how easy Charles has made his recapture, before dispatching only a few ships of soldiers to apprehend him - the colonies would never be able to rustle up more than 5,000 fighting men, probably fewer than 1,000 would be capable of resisting any Commonwealth soldiers in one place at one time, and those men have no military experience at all and could be routed by a force 1/4 of their size. Even if Charles succeeds in evading capture then what life is he living? On the run from a few hundred men, in a land still not safe from Indian bands, living in poverty and humiliation, with probably no more than 10 companions, when he could be living in palaces in France or the Netherlands, able to raise tens of regiments when the time is right. Even without the Commonwealth dispatching troops to arrest him, he's never going to be able to win back England, his name will now be too dirty for Parliament to invite him back on the potential collapse of the Commonwealth, and as soon as England sets its mind to it, the colonies will be reconquered anyway. It's just not a logical move for Charles.
Please don't take this as a dig at you, that's certainly not how it's meant, but I fear this forum is fixated on this idea that Kings were always one coin-toss away from running from the slightest trouble to hide in the colonies and make a new Kingdom there. It just didn't happen like that. It happened to Brazil in 1810-ish because by that point some of Brazil's splendour was reaching that of Portugal's, and the Portuguese royals knew that Napoleon himself was hunting them down and that in a few years all of Europe could have fallen. This is not the situation that other monarchs faced, with far poorer colonies and far less of a desperate situation.
Of course, that said, your TL doesn't need Charles to flee to have the Commonwealth survive. In fact the extermination of the royal line is likely to better produce that end. In addition, in response to your first comment about whether other people have made TLs about the English royals fleeing to America from an overthrowing and forming a new American Kingdom, yes, it's been done a lot of times.