How did the English Civil War and the following reign of Oliver Cromwell affect policies towards and in the American Colonies? I don't see anything Wikipedia that covers this topic (I can only see mainly the militaristic aspects of it there).
I can't offer a good answer, so I'll offer a bad one. *
One of the impacts of the English Civil war was the creation of Whig philosophy - the notion that although England would remain a Monarchy, supreme power lay in the hands of Parliament. (Obvious oversimplification here; it might be more accurate to ascribe this to the Glorious Revolution, but to my mind the two are inseparable).
This notion gave rise to a couple of related notions. (In each of the points below I'm citing the influence of a broader principle on these two events; in no way to I mean to limit these conflicts to this period, but think that the Civil War and the American war for independence represent significant inflection points in the development of these notions.
Taxes were the free gift of the English people (yes, I'm aware that this is an issue larger than the Civil War, but I think there is an inflection point here). This leads to the colonist's "No Taxation without Representation".
Mistrust of a standing army, particularly one controlled by the crown. This has a significant impact on the retirement of military personnel and the quartering of those people in the colonies, which impacts the taxes paid.
Supremacy of Parliament - more than any other conflict the American war for independence is about how Parliament controls colonies. The United Kingdom had colonies, but had no notion how to rule them. In general, landholding was related to representation, but there was no mechanism for the creation of new boroughs. Parliament was still inventing Parliamentary government, and now they had to change the (unwritten) constitution.
* In that I have no sources, just observations.