I was rereading Christopher Hill's often-intriguing book Puritanism and Revolution and came to his chapter on Hobbes. It seems relevant to the discussion here of Locke and of religion in the American Founding.
Hobbes and Locke were contemporaries in adulthood, though Hobbes' writings predate Locke's. Locke certainly was influenced by Hobbes' work. Both men address the question of how to reconcile natural rights, government authority, and religion.
For Hobbes, there was no such thing as natural rights. The idea of a "state of nature" is, as Hill puts it, "a logical abstraction rather than a piece of historical description." For Hobbes, humans without government were humans in chaos; the "natural state" was one of want, war, and ignorance.
Therefore, when nonconformists in Hobbes' day said that a government that did not respect natural law or natural rights could be legitimately overthrown, or at least not obeyed, he responded that this was nonsense. It is society, organized into government, it is government itself that creates all rights and laws, and so there is no way to use some imaginary pre-civilized era as a control over or yardstick for the legitimacy of a human government.
When it is the state itself that creates all rights, then the only way to decide what is just is to have the state decide. This seems like a harsh "might makes right" philosophy, but if you follow it through, it leads to both separation of church and state and religious tolerance. Because politics/government are purely and completely human-made, then religious belief or doctrine has no place in it. We created it, we run it, we make its rules, and we are the final authority over it. Because God is not at all human-made, politics has no place in religion. Humans cannot have authority over God, and therefore humans cannot say which religion is the true religion, and have no authority to persecute anyone for their religious beliefs.
In a democracy, then, the people make their own government and give it the right to decide what is just, and pursue religion privately with no government interference.
Locke, of course, did not agree with Hobbes that there was no natural law, and no natural rights. And it was Locke who appealed to the American Founders, for his philosophy grants our government a sort of spiritual authority, wrapping our human laws and decisions in the mantle of obeying a kind of cosmic justice. This is what makes it easy for people to rename natural law as God's law, specifially Christianity. We say, our laws are rational products of the Enlightenment, but they are also tapping into God's law, the world God made for humans before we started making governments. We're living how God meant us to live.
I think the Founders generally took the view that in creating our democracy they were fulfilling not only their human potential, but restoring cosmic justice.
But they remained a little Hobbesian, too. I think the Founders understood government to be a human creation which is best understood in human terms. And they knew that the authority to decide what was democratic, what provided liberty and justice for all, came from themselves and the citizens of the United States. If it did not, what would be the point? How would the U.S. government be new if it claimed godly justification, just as every government in history had done beforehand?
No, the Founders did not threaten dissenters with God's fury. They took a Hobbesian view that the government they and the people were creating would live or die on human merits, and in doing so raised the bar for what human law, what government, should accomplish.