I asked before about Cromwell, who it seems was viewed negatively. What about the ideas of people like the Levellers, or more general republican ideas that existed in England at that time?
John Adams and Thomas Jefferson went on a tour of the English countryside in 1786. This is what Adams had to say about their visit to Edgehill and Worcester, the sites of two major battles of the English Civil Wars, in his diary:
Edgehill and Worcester were curious and interesting to us, as Scaenes where Freemen had fought for their Rights. The People in the Neighbourhood, appeared so ignorant and careless at Worcester that I was provoked and asked, “And do Englishmen so soon forget the Ground where Liberty was fought for? Tell your Neighbours and your Children that this is holy Ground, much holier than that on which your Churches stand. All England should come in Pilgrimage to this Hill, once a Year.” This animated them, and they seemed much pleased with it. Perhaps their Aukwardness before might arise from their Uncertainty of our Sentiments concerning the Civil Wars.
It should be noted that the battle of Worcester was led by Cromwell himself and marked the final defeat of the Royalists in the Civil Wars. It was fought under the banner of the Commonwealth/Republic, as the monarchy had long been abolished by then.
Another factoid relates to Algernon Sidney, a staunch republican who fought on the parliamentarian side in the Civil War. He initially opposed the execution of Charles I but later endorsed it. He was executed under Charles II for, among other things, writing an anti-monarchist tract. John Adams, who was far from a radical, thought very highly of Sidney and his writings. Some even claim Sidney's Discourses on Government was more influential among the Founders than the works of John Locke.
So I think it's safe to say the American Founders' sympathies lay firmly with the parliamentarian side in the English Civil Wars, including the republican period after the monarchy had been abolished. They may not have been as radical as the Levelers or supporters of Cromwell, but they were not mere English Whigs tracing their political lineage to the Glorious Revolution either.