An interesting question, and one that is perhaps insufficiently studied in US history.

The Puritan colonies of New England were supporters of the Parliamentary and Puritan side in the English Civil war. They had come to America to build a new Puritan society without the inconvenience of all the non-Puritans to be found in England. Now it appeared possible to build a Puritan society in England, rather than in the howling wilderness.

Many New Englanders returned to England to fight for the Parliamentary cause. An estimated 7-11% of the then 40,000 population living in the colonies which would later become the U.S. returned to England. This was the largest (proportional) emigration from America in history.

https://www.boundless.com/u-s-history/textbooks/boundless-u-s-history-textbook/britain-and-the-settling-of-the-colonies-1600-1750-3/settling-new-england-41/puritanism-277-10429/

I believe that the eventual outcome of the English Civil War (Cromwell's dictatorship and then the eventual restoration of the Monarchy) was very disturbing to Puritan New Englanders. How this experience shaped the thinking of New England would be interesting to research.

Other colonies in North America (Virginia in particular) seem to have supported the Royalist cause.

You can find other responses to this question in this previous link:

http://www.reddit.com/r/AskHistorians/comments/12xdmq/what_happened_to_the_new_world_colonies_of/

and, I believe, there is also a section in the FAQ which might have more information.