How did it impact immigration rates and immigration demographics to the colonies from the British Isles? Were there political controversies in the colonies about which side to take? What were the sentiments in the different colonies regarding Cromwell? And did any of the other colonial powers try to exploit this period, in terms of seizing territories in the new world while the English were preoccupied?
This earlier post might be of interest to you, though it doesn't answer all your questions.