A difficulty here is sorting out the theological legacy from the political one. Puritanism was on the go long before the political problems of a king who thought he had a divine right to rule and that his will was equivalent to God's will resulted in political upheaval in Britain and a civil war in England.
From even before the time of Constantine, the name of Cathars, or Puritans, was applied to groups of Christians who had objections to bending their knees to popes or to departing from biblical Christianity. There have always been such groups throughout the centuries, known by various names, but linked by their determination to uphold the purity of scripture and to take that as their rule of faith. They were hounded from pillar to post, many dying horrible deaths.
During Queen Mary Tudor's rule (1553-58) she killed so many Protestants, there was an exodus to America, to flee her bloody reign. That's when the Pilgrim Fathers established Puritanism in America.
By the time of Queen Elizabeth the 1st, a group remaining within the new Church of England had consistently striven against what was Romish in it, endeavouring to make it more definitely Protestant. Those Presbyterians were more in sympathy with the Continental Reformers than the Church of England was. Presbyterianism was established in Scotland, but the authorities in England would not tolerate it. Those Independents were crowded into prisons, fined, mutilated and executed with unrelenting cruelty. The Baptists were looked upon as even worse because they claimed the state had no authority to interfere in matters of religion. Although the Puritan group in England maintained political activity to be vital and so were at odds with the Baptists on that point, they had many similar theological beliefs in common.
The Puritans pushed for a better translation of the Bible in 1604 and King James 1st sponsored the production of the Authorized Version of the Bible of 1611. Around that time, Oliver Cromwell was being educated by a Puritan schoolmaster and at a Puritan college. But it was not till 1628 that he was converted to faith in Christ. He was an Independent by church affiliation but gave room to others. He was elected an MP in 1629 and his first speech was a protest against lack of religious freedom. King James 1st responded by dissolving Parliament and ruling absolutely for 11years. In 1638 James 1st tried to impose a Book of Common Prayer on the Scots but war followed and he had to concede, reconvening Parliament. To his dismay Parliament rallied to the Scottish side so he dismissed Parliament again. But when the Scots invaded England, he had to recall Parliament in summer 1640.
This 'Long Parliament' remonstrated with James 1st (for reigning 11 years without Parliament). The English Civil War stared on 22 August 1642, Oliver Cromwell becoming leader. He signed the death warrant of the king, January 1649. That was when monarchy was abolished and the Commonwealth appointed Cromwell as Lord Protector in December 1653. He was offered the crown on several occasions but always refused. He died in 1658 but two years later General Monck marched down from Scotland and the monarchy, with Charles 2nd as king, was restored.
History homes in on the awful things that happened during the Civil War, blaming "the Puritans" in general, and Cromwell in particular (especially for banning Christmas in 1651!) Yet the Puritans and Cromwell established democracy! Parliament's authority was secured. And Cromwell established religious freedom so that when Charles 2nd tried to force religious uniformity in Britain, 2,000 ministers rebelled, many at great personal cost. They had learned freedom under Cromwell and chose not to give it up. Although Cromwell had been intolerant towards some forms of religion, compared with most, he was a champion of freedom.
Now, I give that brief summary to show how complex the situation was in Britain and why the Puritans, and Cromwell, have been inextricably linked with all the political horrors of that era. Yet if there had not been the cruel persecution of Puritans by Queen B....y Mary and the dictatorship of King Charles 1st, there might not have been any Civil War in England. In every era, Christians have to respond, in faith, to prevailing conditions. But they more they get mired into politics, the harder it is to remain true to Christ. the Puritans did what they felt best for the nation, and to not lose the religious freedom gained. Had they stayed out of politics, King James 1st would have lived on and there would have been no Parliament in Britain, as we know it. The divine right of kings idea he promoted would have been perpetuated, instead of democracy. There would have been no King James Version of the Bible. Of course, many bad things happened too - yet how can we judge them, so far removed from the times of that era? People were forced by circumstances to take a stand, both religiously and politically, not least because of the evils of dictatorial kings and queens. Yet because of the Puritans and Cromwell, we no longer have to endure that. The evils that still go on in politics can yet be curtailed by the voting of us who benefit from democracy. We have the Puritans to thank for that, both in Britain and in America.