One aspect of seventeenth century England I have understood is that it had an increasingly fractured religious and political environment, with a wide array of strikingly radical political groups, such as the Levellers, heterodox religious movements such as the Adamites, and both such as the Diggers. As far as I know these groups were uniformly on the side of the Parliamentarians, but were they able to exert any influence over the Commonwealth? All I really know about Cromwell's social policy is that he banned Christmas.
(Repost of previously unanswered question)
Good surveys of this period are: Christopher Hill's The Century of Revolution and The World Turned Upside Down; and Mark Kishlanskey's A Monarchy Transformed Britain 1603-1714.
Regarding the Levellers; Joseph Franks' The Levellers, HN Brailsford's The Levellers and the English Revolution and Pauline Gregg's Free-Borrn John are the standard works. I think all of the Leveller books are currently out of print but they were in print in the 1960s and used copies can usually be found on line.
The consensus of British historical opinion holds that the Levellers exerted an enormous influence on the revolution between 1646-49 but their light was extinguished after 1650. The Levellers drove Cromwell's Independent faction in Parliament far to the populist right or nationalist end of the prevailing liberal and republican spectrum of political opinion in the Long Parliament of the 1640s. They had enormous influence in the New Model Army's Army Council and their greatest achievements were the Putney Debates in October and November 1647 and the several Agreements of the People that were published over the following two years. Henry Ireton's Heads of Proposals was a response to the "Agreements* and the Heads ultimately became the Commonwealth's Instrument of Government.
The British marxist historians agree that ideas and opinions of the Levellers are easily detected in the American, French and Russian revolutions but they also agree that there is little evidence that any of these revolutions were directly inspired by the Levellers.
However, in a footnote in The Ideological Origins of the American Revolution, Bernard Bailyn says that Thomas Paine's "Common Sense" can be traced to Leveller pamphlets of the 1640s-50s and not to Locke or the Enlightenment. This is supported by the observation that the New England militia rose in April 1775 and had been in the field besieging the British in Boston since June 1775 but that Paine's Common Sense was not published until January 1776. This strongly suggests that Paine didn't change anyone's mind, he was merely preaching to the already revolutionary choir.
In my opinion, the Massachusetts Bay Colony, the New Model Army and the Levellers were, as a good Puritan would say, the "first fruits" of the Puritan Revolution of 1628-60. The Puritan colonization of the New England colonies began in 1628 and the New Model Army and the Leveller movement were formed in 1645. All of them drew their members from the same religious and socio-economic class; i.e. reformed, Independent Protestant yeomen, husbandmen, apprentices and artisans. They all exhibited the same radical republicanism in the context of Independent Puritanism.
In the 1640s, the two grand divisions amongst the Parliamentarians were the Presbyterian faction and the Independent faction. Theologically, they were, and remain, indistinguishable. They differed on the method of church government. The Presbyterians wanted a Whiggish sort of hierarchical system while the radically republican Independents thought each church should be both independent and self-governing within a very broad definition of acceptable calvinist theology. In the context of an established church, this had enormous political consequences. These consequences were fully played out in New England between 1650-1704.
After 1647, the major divisions of the Long Parliament in the popular mind were Independent and Leveller; Presbyterian and Grandee and Royalist and Cavalier. These distinctions are very crude but reasonably accurate.