The answer is there is no clear cut answer; we probably will never know. When I wrote my "briefly noted" article for First Things on James H. Hutson's quote book on the Founding & Religion I stated:
While all the Founders believed in a powerful Providence, there was a split between those who affirmed the tenets of traditional orthodox Christianity and those who subscribed to an Enlightenment-influenced "theistic rationalism." While orthodox Christianity dominated the views of the population at large and probably a statistical majority of those who signed the Declaration and framed the Constitution, an unconventional Unitarian theology seemed to engage the minds of certain key Founders—among them, those who played the most prominent roles in declaring independence and drafting the Constitution.
Were I to write another piece on the matter, I might use less strong words than "orthodox Christianity dominated the views of the population at large...." It's possible that most of the population were "orthodox Christians." It's likely that most were somewhat affiliated with a Christian system that professed "orthodoxy" and they didn't challenge said theological tenets. The more I think about it, however, the more I doubt that a statistical majority of Americans during the Founding era actually believed in things like Trinity, Incarnation, Atonement, and the infallibility of the Bible (i.e., "orthodoxy"). They might have; however, the record is just not clear that they did. The record IS clear that almost everyone from that era believed in Providence.
One notable study from that era showed that ONLY 17% were members of a church. That Founding era Americans were more likely to be in Taverns on Saturday nights than in Church pews on Sunday mornings. Other evidence shows that this may be a low ball. However the bottom line is that we just don't know whether a statistical majority of Founding era Americans accepted such theological tenets as Trinity, Incarnation, Atonement, etc.
John Derbyshire once notably said something like "the lazy Christian mind is reflexively Deist." Indeed, evangelicals should understand this given that their faith stresses the "narrow gate." Roger Williams, a fervent evangelical-fundamentalist, interestingly enough, understood this dynamic and used it as a cornerstone for arguing in favor of separation of Church and State and religious liberty. Williams argued the inevitable not only existence but perhaps statistical majority of the "unregenerate" in any given population of "professing Christians" makes the idea of a "Christian Nation" blasphemous.
“Deism” as a significant theological conversation ended at the end of America's Founding era. However as a theological “reality” — something in which nominal Christians believe — I think various kinds of deism and unitarianism are not only alive and well today, but probably have always been, again perhaps always dominated "Christendom."
As Jefferson himself put it:
I remember to have heard Dr. Priestley say, that if all England would candidly examine themselves, and confess, they would find that Unitarianism was really the religion of all;…
– Thomas Jefferson to John Adams, August 22, 1813.
Forms of Deism and Unitarianism tried to give an intellectual account of this reflexive, default position into which nominal Christians fall. I can’t tell you how many professing Christians I speak with today — folks who haven’t spent too much time thinking about these issues — who believe God exists, that He wants humans to do good to other humans, that good people get into Heaven — but also that have no strong belief on matters like Trinity, Incarnation, Atonement, and infallibility of the Bible. A little while ago a Christian source did a story on this calling it a “new” religion of younger Americans. I noted that there was nothing “new” about this creed. Since the time of the American Founding it has arguably been the dominant creed, the “broad” gate, as opposed to the evangelicals’ “narrow” gate.
Why is this relevant: In arguing over America's Founding political theology, I oft-hear that we shouldn't focus on a "top down" view of things (i.e., the Christian-Deists/Unitarians/Theistic Rationalists elite "key Founders") but rather a "bottom up" view of things (i.e., the "orthodox" masses). Well, it's not clear that a statistical majority of Americans during the Founding era really were "orthodox Christians," but rather were nominal Christians who, if they really "candidly examine[d] themselves" would profess a creed something closer to Jefferson, Priestley, the "key Founders" than orthodox Christianity.