So a major problem is how to get the king across the sea when there's nothing across the sea to attract him. Here's a crazy-ass solution: Suppose Charles I is not executed. Instead he is sent into exile in Boston, where he is supposed to be guarded by zealous and incorruptible puritans and kept far away from any royalist conspirators. But his nephew Rupert, posing as a whaler, manages to smuggle him south to the friendlier shores of Maryland. Here he is reunited with supporters and sets up a temporary court. He spends years planning his return to England, but his long absence allows parliament and the Anglican church to arrange a "constitutional restoration", awarding the crown to a suitable Protestant.
Charles remains in the Chesapeake, where he is joined by his sons and others who support his claim. After his death, his son Charles II is crowned in Baltimore. He spends his entire reign in Virginia and Maryland, working to fortify his navy and secure the borders of the "true dominion". He forges alliances with Spain and France, signs treaties with the local tribes, and constructs a permanent capital, Kingston, at the mouth of the Severn River. His brother James succeeds him and continues his work.
The Stuart dominion, known generally as "Virginia", comes to dominate the Caribbean and the middle colonies, winning decisive naval victories at Bermuda and Long Island. In 1750, Virginia and England sign a treaty whereby England recognizes Virginian sovereignty over American territories south of the 40th parallel, and the King of Virginia renounces his claim to the crowns of England, Scotland, and Ireland.