Well, I can tell you that the 13 Colonies pretty much constituted a microcosm of interregnum era England. New England settlers largely came from England's puritan cities, such as Cambridge, Boston and Hatfield (which many New England places are named after), and came to support Cromwell's Commonwealth as a result. Many American settlers actually came back to England to fight for the Parliamentarians. Half of all Harvard graduates in the 1640's in fact ended up fighting on the Roundhead side of the War. Virginia, on the other hand, formally a royal colony, was settled by devout Anglicans and admirers of England's great rural estates, and naturally supported the King. Battles were fought between the Roundheads and the Cavaliers in North America, such as the Battle of the Severn, considered by some to be the last battle of the English Civil War. Following Parliament's victory, Royalist sentiments lived on in the colonies. Virginia attempted to resist Cromwell's economic and imperial policies, maintaining that Charles II was King. Cromwell dispatched a fleet and accepted the surrender of Virginia without a shot being fired. Hope that answers your question.

As an aside, when talking about the 1650s and the time of the Interregnum, it is anachronistic to talk of 'thirteen colonies'.

In this time, what would become New York, New Jersey, Pennsylvania and Delaware were all under the control of Dutch. England would gain control of these regions in the 1660s, after the restoration of Charles II.

Additionally, settlements in the Carolinas would first be established in the 1660s after the Restoration, and Georgia was not established until 1732.

In the 1650s, the major English colonies in the Americas were those in New England (Massachusetts Bay, Plymouth, Rhode Island, Connecticut, New Haven) Virginia, Maryland, Newfoundland, Bermuda, several islands in the Lesser Antilles (St Christopher, Nevis, Barbados, Antigua, Montserrat, Trinidad, Tobago). Additionally, an English Commonwealth navy invaded and took Jamaica from Spain in 1655.

/u/Haakon_VII gives a good synopsis for the attitudes of colonists in New England and Virginia. However, in some colonies like St Christopher, Barbados, and Antigua, governors and the island populations were more ambiguous in their loyalties, and representatives from the Royalist and Parliamentarian causes lobbied these governors to support their respective sides.

For a good book about these various colonial viewpoints on the English Civil War, I recommend The English Atlantic in an Age of Revolution; 1640-1661 by Carla Gardina Pestana.