This is a big question, I won't be able to do it justice, but here's a start. In short, yes, there are significant differences in religious life between England and the N. American colonies and these differences come into play during the Revolutionary War. To this day Queen Elizabeth II is Supreme Governor of the Church of England and Defender of the Faith whereas the United States famously affirms the First Amendment.

The main difference you need to appreciate is between the Church of England and dissenting Christians. If we go way back to Henry VIII's Reformation we can see that initially he didn't change theology from the Catholic beliefs, for instance in the 1530s transubstantiation is affirmed, but he replaced the primacy of the Pope with himself. Reformed theologies are spreading in England throughout the reign of Edward VI, then Mary's Catholic repression martyrs Protestants and confirms Protestant theology within Anglicanism. When Elizabeth becomes Queen she's confronted with the problem of reforming the Anglican church to fully establish its Protestant character, like Protestantism itself this is a matter of great controversy, and some conferences of scholars are called to produce what's known as the "Elizabethan settlement." The settlement makes some Protestant liturgical reforms but claims for Anglicanism a magisterial/state authority similar to Catholicism. The Act of Uniformity and Book of Common Prayer are legally mandated, there's no freedom of conscience for someone who is more radically Protestant than the C of E.

And over the course of Elizabeth's reign the ranks of those radical Protestants are swelling! Called Puritans in England, these are people who refuse to attend the Anglican Church and who acknowledge no authority above the Scripture interpreted with aid of Grace. I've written some big comments on Puritanism if you're interested.

Between 1600-1650 there is growing conflict between these dissenting Protestants and the Anglican Church, many emigrate to the New World in search of greater freedom to govern their religious affairs. It's important to note that these groups want to govern, they want not only freedom from persecution but the freedom to persecute! In the 1640s England has a Civil War between Parliament and King, Parliament being dominated completely by dissenters. Oliver Cromwell is himself a (relatively mild) religious independent.

So from the beginning of settlement the American colonies have significant religious diversity and this is one of the factors encouraging emigration. It varies regionally, but some areas are quite mixed, there's plenty of conflict. In 1689 there's even an anti-Catholic coup in Maryland.

Dissenters remain in Britain too, of course, but they are henceforth always a minority.

About Congregationalism more specifically. Some radical Protestant theologies are associated, fairly or unfairly, with specific political ideas. Cromwell's revolution had a religious character, they killed a King, and there's a persistent tradition of anti-authoritarianism in some of these groups. Quakers would be the best example. "Congregational" refers to a mode of Church government different from the CoE and implictly Calvinist. Indeed the historic Congregational churches were strongly Calvinist. In 1776 a British person would connote Congregationalism with anti-Monarchical ideas/sentiment, since the Revolutionary War was framed by loyalists as being about loyalty to a legitimate King you can imagine why a Congregational church would attract abuse.