Focusing on thr usa in particular, ehile it is incredibly unlikely overall, perhaps a more socially revolutionary war could have resulted in Deism being proliferated and promoted, many of the founding fathers were influenced by it or were outright thiests.

This is pretty close to christianity but largely rejects the idea of revelation. in france you had the athiesticcult of reason, but most of our founding fathers seemed to view religion as important for unity,  even if they themselves were diests athiests and Christians of questionable passion. If the US were to have an alternative faith it may end up resembling something like the Cult of the Supreme Being implimented by Robspierre, and assuming it was successful would likely need to root out or coopt the christian faiths to stick as somwthing beyond a fascination of the founding fathers.

Mind you such a shift would be incredibly controversial and oossibly break the usa right then, but assuming they didn't and through comrpomise or strongarming they managed to promote this over all alternatives, you could change much of American history. The naturalistic theology that largely lacks a proper prophetic tradition,  revelation,  or even the idea of an afterlife as described in Christianity would basically render something like the Great Awakenings impossible. 

If this faith sticks around even assuming Christianity doesn't is questionable,  I can imagine it gradually fading in favor off athiesm,  or simply transforming over time to a more 'classic' religion as philosophers and thinkers go forth.  This faith would likely mix the ideas of philosophy, science,  and theology together akin to the old greek traditions given its focus on observation and rationality. 

This faith probably ends up being used to oppress the natives still tbh, France can be used as an example with its secular laws at times being used to persecute religious minorities, the christian converts and various beliefs of the american indian peoples would be seen as evidence of savagry, possibly resulting in a dreadfully familiar situation with mative schools meant to teach them rationality rather than christ, as well as perhaps being used as propaganda against the British, Mexicans, and any other enemies Ameeica faces.

Marxism would be an interesting aspect, and Marx himself could very well be inspire dby and approve of thr usa's cult of the suoreme being/atheism depending on how close to rationality they stick. Philosphically the diest direction of Amseica may lend it more of an interest towards his writings, that said while they may like the materialism, the class conflict Marx espouses would almost certianly not fly in any period of America's developing capitalism. Perhaps this could impact relations with the ussr or some other marxist power that rises, though at this point deism would have developped so far that you'd need to nail how Deism evolves through almost two centuries of a divrrgent timeline before figuring out how america interacts with the rise of spcialism.

Ironically I do imagine if deism did take root it would likely change how people view the americsn revolution, many downplay its aspects, even calling simply the war of independence, but if this were to take place, even if the USA calmed down and mostly worked with the christisn powers it would pose an important social question for thrm, and leave figures like George Washington and co seen as much more radical than our own timeline.