Arda has all of Earth's religions by definition, just fast forward a couple Ages. The Athrabeth Finrod ah Andreth sort of posits a Fall of Man - divine incarnate scenario when it discusses Eru's original plan for mankind (it also, interestingly, states that the first Music ended for the Ainur just as the Firstborn began to fade from Middle Earth e.g. around the early Fourth Age, which means anything after, including our history and the transitional period between Middle Earth and prehistorical Earth societies, is unknown to the Valar). It's merely a hypothetical, but it's quite interesting in the context of Tolkien's views on Mannish beliefs and Messianism in the aftermath of Melkor's corruption.
'Truly,' said Andreth. 'So may Eru in that mode be present in Eä
that proceeded from Him. But they speak of Eru Himself entering into
Arda, and that is a thing wholly different. How could He the greater do
this? Would it not shatter Arda, or
indeed all Ea?"
'Ask me not,' said Finrod. 'These things are beyond the compass of
the wisdom of the Eldar, or of the Valar maybe. But I doubt that our words
may mislead us, and that when you say "greater" you think of the
dimensions of Arda, in which the
greater vessel may not be contained in
the less.
'But such words may not be used of the Measureless. If Eru wished
to do this, I do not doubt that He would find a way, though I cannot
foresee it. For, as it seems to me, even if He in Himself were to enter in, He
must still remain also as He is: the Author without. And yet, Andreth, to
speak with humility, I cannot conceive how else this healing could be
achieved. Since Eru will surely not suffer Melkor to turn the world to his
own will and to triumph in the end. Yet there is no power conceivable
greater than Melkor save Eru only. Therefore Eru, if He will not relinquish
His work to Melkor, who must else proceed
to mastery, ``then Eru must
come in to conquer him.``
'More: even if Melkor (or the Morgoth that he has become) could in
any way be thrown down or thrust from Arda, still his Shadow would
remain, and the evil that he has wrought and sown as a seed would wax
and multiply. And ``if any remedy for this is to be found, ere all is ended,
any new light to oppose the shadow, or any medicine for the wounds: then
it must, I deem, come from without``.'
The Blue Wizards may or may not have seeded the basis for Eastern mythology too.
'I really do not know anything clearly about the other two [wizards] – since they do not concern the history of the N[orth].W[est]. I think they went as emissaries to distant regions, East and South, far out of Númenórean range: missionaries to 'enemy-occupied' lands, as it were. What success they had I do not know; but I fear that they failed, as Saruman did, though doubtless in different ways; ``and I suspect they were founders or beginners of secret cults and 'magic' traditions that outlasted the fall of Sauron.``' - J. R. R.
We have no idea nor will we ever have an idea what the peoples of Harad believed in. Sauron/Melkor worship scattered in the north, probably, but Far Harad is one big blank. Ditto for Hyarmenor or the New Lands that replaced Aman after the Breaking of the World or the peoples that inhabited the Straits of the World (Mesopotamia) or either First Age and Third Age incarnations of Dorwinion (though they were probably Eru worshipers of some form considering they traded regularly with the western peoples of Middle Earth).