Hey there. I actually answered similar questions not long ago, so I'll be pulling my answer mostly from this one and from this.
So, to answer the primary part of your question, during the American Revolutionary period (1770 - 1790), the average American was not religious in any visible way. By religious, I simply mean, if those people took part in common religious events like attending church regularly, which most did not. According to most historians, only 15% of Americans attended church once a year or more by 1775. (1) Some historians, like Patricia Bonomi, have argued the number was likely higher, and estimates it to be about 30%. Regardless, church attendence was not super common place during this period.
The founders were similar. Some founders, like John Adams did attend church regularly, however Adams was actually a proponent of separating church and state. Other notable founders, like Ben Franklin and Jefferson were most notably not Christian. Even some founders that did attend church regularly, like George Washington, were believed to do so for appearances. In his biography of George Washington, Ron Chernow explains:
Many mysteries have surrounded George Washington's religious beliefs. A pair of notable acquaintances raised questions about his faith. Thomas Jefferson once remarked cynically that Washington, "has divines [ministers] constantly about him because he thinks it right to keep up appearances but is an unbeliever." Jefferson contended that when Washington stepped down as president, a group of clergy-men presented him wit haha list of requests to bolster public faith in Christianity; they noted he had refrained from public endorsements of the tenants of christianity and beseeched him to declare openly his beliefs. According to Jefferson, "the old fox was too conning" for the preachers and replied to all their points except the one about his personal faith... Bishop William White of Pennsylvania, Washington's pastor during his presidency in Philadelphia, also stated, "I do not believe that any degree of recollection will being to mind any fact which would prove General Washington to have been a believer in Christian revelation."
This is interesting because it shows that while Americans themselves were largely irreligious, it was believed that going to church, at least for public figures, ensured some public sense of good virtue. Some groups, like Methodists and Quakers were more religious and did attend church with regularity, but these were largely minority religions by this period.
I would also like to make it clear that while the majority of Americans during this period did not go to church or do other religious customs, the overwhelming majority of Americans would self - identify as Christian. Practically all states had restrictions against non-Christians from voting or running for public office, so identifying as one, even if just in name, had societal bonuses.
So what changed? Well, a lot changed. Between 1790 - 1820, a period commonly known as the Second Great Awakening happened in America. This period of religious revivalism sought not only to make America more Christian, but also re-write popular memories of the period to make America's founding seem more religious than it was.
The tone of Religion began to change in 1790, but really started to ramp up in the 1810s. Jon Butler, a leading American religious historian repeatedly argued that the reason we believe today that the earliest generations of Americans were highly religious, and in turn started to claim we were a "Christian Nation" is because later generations desired that we believe it. Butler argues that religion was used as a tool for the religiosity of later generations who were exaggerating and even fabricating some of their claims about America’s religious past to spread their own messages. This can clearly be seen in many of his works, including his book Awash in the Sea of Faith: Christianizing the American People. In this book, Butler also explains that America’s religious leaders feared this secularization, and sought to bring back religion into American society. Here Butler explains:
Between the signing of the Declaration of Independence in 1776 and George Washington’s death in 1799, American church and denominational leaders renewed efforts to stamp Christian values and goals on a now independent society. Three of these attempts proved especially important: powerful Christian explanations of the Revolution and of the proper political order that ought to govern American society, attacks on irreligion, especially on skepticism and deism, and the creation of new religious groups, which evinced principles that for the first time might be called distinctively American. (pp 212)
This push towards having a religious revival was not orchestrated by united leaders of different Christian denominations, but rather happened independently that road the waves of change in American society during this period and can be seen when examining independent religions. Methodists, which were a minority religion in 1776, boomed to having 250,000 members by 1820, and by 1830, that number doubled. (7) Similar themes for Bapists happened, where their numbers tripled between 1783 and 1813.(3) Part of why this was so effective is that Christianity became an effective enterprise where being a part of a growing church was highly desirable. Part of this revival also intertwined from the desire to repaint America’s origin stories as being extremely virtuous and Christian. An example of this can be seen through biographies written on the founders by American pastors and ministers. Mason Weems, for example, invented the story of George Washington being unable to tell a lie about chopping down his father’s cherry tree. The story appears in his fifth edition of Washington’s biography, and is certainly a figment of his imagination, however it was at the beginning of a trend that sought to paint the founders as deeply moral men that would never do anything that lacked virtue. This is especially true for other founders who held deistic beliefs, which was harshly frowned upon by the 1810s.
Ultimately, the founders sought to instill secular values into the American government when support of organized religion was comparatively low. When religious revivalism spread through America during the ensuing decades, people sought to portray America’s religious history as being deeply rooted in Christianity, although the sources do not suggest this is true.
Edit: Added a contextualized paragraph.