With Jamestown in 1605 and the Plymouth colony established by 1620, I can't imagine the relatively new and scattered settlements weren't effected by three civil wars ending in, what could be called, a military dictator controlling the motherland less than years later.
Given the religious motivations behind some settlers and religious seeds in the conflict itself, the politics seems like it would be relevant to early American colonists.
From an earlier answer of mine:
In 1640, England’s Colonial Atlantic empire spanned twenty-four significant settlements consisting of around fifty thousand people. Most of these colonists had been born in England. However, the older colonies were home to a generation born in the New World, while there were minority populations drawn from the other nations of the British Isles. On paper, English territorial claims stretched from the Caribbean to Newfoundland. Each of the colonies was part of an interdependent network, exchanging not just goods and supplies, but ideas, religion, and people. The English colonies had been settled over the previous thirty years. The most populous was Massachusetts, followed closely by Barbados, then Virginia, with the others coming in between a few hundred and a few thousand settlers by 1640.
In 1640, the colonies were highly connected to England, politically, ideologically, and economically. Politically, the colonies were usually administered by an England-based charter company, or were the personal assets of an England-based proprietor. For example, the proprietor of the Maryland colony was Cecil Calvert, the second Lord Baltimore, while the earls of Warwick and Carlisle both held significant proprietary rights over many of the colonies. These proprietors received their grants from the king, who himself administered the Crown Colony of Virginia. Outside of these groups, there were a number of New England colonies which had no official permission to exist from England, the “independent” colonies of Rhode Island, New Hampshire, New Haven, and Long Island. These colonies backed up their existence with agreements with Natives and other colonies, but their positions were far from secure.
In religious matters, the colonies were meant to conform to Church of England practices, but they rarely did so. Much of New England, for example, was ideologically opposed to the reforms of the Archbishop of Canterbury, William Laud. For most other plantations the distance from authority and the precarious situation meant that religious conformity was hardly a priority. When you’re struggling to survive, you aren’t going to be that fussed about whether your minister is wearing the right clothes! Harvard College, founded in 1636, produced a legion of ministers who were snapped up by colonial governments, desperate for any trained preachers they could find. That these men would likely have fallen foul of official Church policy back in England was carefully ignored.
Whether they reported and paid taxes to the king, to a noble proprietor, or to a company, the colonies were far from subservient. Colonists could, and did, object to the governors imposed from above, and by 1640, more than half had established their own legislative bodies, with even Charles accepting the existence of the Virginian House of Burgesses. There wasn’t universal success in this, however, and Newfoundland and Maine, among others, were prevented from forming their own legislatures by particularly strong English authorities. However, whichever colony they were in, almost all colonists had been born in England, lived under English law, tried to recreate English institutions, and considered themselves subjects of Charles, whether they agreed with his policies or not. News from England was eagerly dispersed throughout the colonies, whether it arrived by letter or by settler, especially after war erupted back home.
First Charles’ subjects in Scotland, then in Ireland, and finally in England took up arms for and against him. Once fighting began in England, both King and Parliament expected the colonies to pick a side, and both demanded that the side be theirs. Both King and Parliament would be disappointed.
Let’s look at these expectations first. Everyone, whether they were parliamentarian or royalist or something in between, expected New England to back parliament. Parliament had long believed that basically all the colonists who had left since Charles came to the throne had left because they opposed the King. New England was just the most vocal about it. They had clearly fled his tyranny, and so would jump at the chance to support parliament’s struggle against it. Obviously Charles didn’t think England’s Atlantic world was full of potential rebels, but he did at least consider New England to be a lost cause - it was no secret that they shared many religious and political ideals with his parliamentary enemies. As a crown colony, both sides expected that Virginia would be royalist. For almost all the other colonies, both sides believed that they would respond to the orders of their England-based company or proprietor.
So how were these expectations disappointed? Well, first off, none of the colonies were at all interested in risking their very existence. They had opinions and sympathies with one side or the other, of course, and the leadership of some colonies was explicit in their words of support for either the king or parliament. But that was more or less the extent of official colonial contribution to the First and Second Civil Wars – words, and words are wind. Virginia and most of New England did declare their expected allegiance, as did other colonies, but that was about it. Colonial governments did their very best to make themselves small and hard to notice as the big kids fought in Europe, because you have to be realistic about these things. Even the largest colonies were relatively vulnerable, and vulnerable from almost every direction – none would be able to stand up to a concerted attack from whichever side was victorious, and being too actively engaged in the fighting not only weakened a colony’s defences and risked retaliation, but opened them up to attack from colonies on the opposing side, their Native neighbours, and other European outposts. Massachusetts Bay, Plymouth, New Haven and Connecticut formed an allied confederation explicitly based on fear of attack from other Europeans and Indians, 'by reason of those sad distaccons in England which they have heard of, and by which they know we are hindred, from that humble way of seekeing advise, or reaping those comfortable fruit of protection which at other tymes we might well expecte.'
In 1644, Virginia would come under attack from the Powhatan Confederacy for the third time, who were well aware that the Virginians wouldn’t be getting any support while civil war raged. When Lord Baltimore, proprietor of Maryland, took a privateering commission, his own colonial government opposed him – they were understandably concerned that he was going to sink the ships that were coming to buy their goods and sell them supplies. For the same reason, Massachusetts requested that both sides consider its harbour neutral ground – trading was much harder when the merchants were shooting at each other.
All of the colonies suffered the same divisions between royalism and parliamentarianism that England faced. In Plymouth Colony, a settler challenged the governor to explain why he was not following the king’s orders, and in Massachusetts a ship's captain questioned whether parliament were committing treason. Another challenged a warrant because it had been issued in the king’s name. In Virginia wild rumours spread that the Parliamentarian minority had informed the Powhatan of the civil war and urged them to attack the Crown Colony while England was distracted. Civil unrest was just as dangerous to a colony’s survival as outside intervention. In the First and Second Civil Wars, regardless of a colony’s official position, colonial governments cracked down, not on those agreeing with the “wrong” politics, but on those being too zealous one way or the other and disturbing the peace. The widespread neutrality of the colonial governments wasn’t down to disinterest, but self-preservation.