The only evidence of religious ritual in The Lord of the Rings is when Faramir and his men stand and face West in silence before eating, as an act of reverence to Numenor.
Elsewhere Tolkien makes clear that the Numenoreans in exile (in Arnor and Gondor) have ceased to practice their religion due to the destruction of the Holy Mountain Meneltarma which was the site of communal annual worship, led by the King (everyone else being silent).
The presumption is that until 'The Temple' is (somehow) restored, or a replacement divinely ordained - then the Numenoreans can do more than reiterate their reverence for lost Numenor and via this, the Valar and the One above all.
This is of a piece with the way in which religion works within the pre-incarnational world of the Lord of the Rings - by reverence of that which is higher - usually only a step or two higher - rather than direct reverence of God.
The Good characters among Men (including hobbits) are distinguished by their love of - or at least respect for - elves; which are a higher form of humans, and of The Wise (Wizards, High Elves and the half-Elven).
It is goes down the scale. The non-Numenorean 'Middle' Men of Rohan are ennobled by their treaty with Gondor - apart from that they are 'merely' courageous and loyal barbarians: it is their treaty with Gondor, and via Gondor (but not directly) with reverence of the Valar and the One, that Rohan is lifted above the wild men such as the Easterlings and Dunlanders.
Rohan is above the hunter gatherers - Druidain - but at the time of the War of the Ring this relationship has been broken and is restored during the course of the book. The reverence the Druidain ought to have for the Men of Rohan has been broken precisely because the Rohirrim have come to regard the Druidian as sub-human and denied their duty of care towards them (apparently hunting them as if they were beasts) - similarly Gondor seems to have neglected their duty of care (their noblesse oblige) toward Rohan - allowing them to become corrupted by Saruman.
It seems that Rohan's Goodness is almost wholly mediated by their relationship with Gonder - in and of themselves, the Men of Rohan do not know about elves and the Wise, do not seem to know about the valar or the One. Their Goodness is therefore a function of their reverence for Gondor.
Similarly, Hobbits are corrupted by a selfish and short sighted complacency at the start of the tale, as they have lost knowledge and reverence for 'higher things' - especially elves.
It is precisely those hobbits who are unusual in their respect for elves which save the Shire at the end of the story - left to themselves the insular and comfort loving hobbits have let themselves be taken over by 'ruffians' and lack the psychological resources to resist evil.
Goodness in LotR is substantially about recognizing and reverencing that which is higher than oneself; and this applies all the way up and down the scale - it is equally vital that Elrond (who is acknowledged by all the good characters as leader of the Good forces in Middle Earth) reverence and is humble towards the Valar and The One - as that hobbits respect and are humble towards elves (like Frodo and Sam, and in contrast with Gollum).
The responses to Galadriel are likewise an index of the goodness of characters - even Eomer, who initially makes some rather paranoid and insulting comments about the Lady of the Woods, swiftly backs down and defers to those who know better; Faramir regards elvish contact as too perilous for modern men, but his profound respect for and deference toward the High elves is implict.
Equally the more-noble must have responsibility and care for those below them; as Gandalf does and Saruman does not; as Theoden lost and then regained, as Denethor lost and never regained.
Without this paternalism, Theoden and Denethor were unworthy to lead, and needed to be removed from responsibility - whether temporarily (Theoden) or permanently (Denethor).
Tolkien's point is a characteristically Catholic one: that simple folk can only have a right relationship with the highest things by means of intermediate things - and that we are all of us (or almost all of us) in this respect simple folk - we can only comprehend that which is a step or two above us.
In his personal devoutness it seems that Tolkien's reverence for God was primarily via 'intermediaries' such as priests, saints, angels and the Blessed Virgin Mary - and only indirectly by Jesus Christ who he perhaps saw as far above him - too far above him to have the direct and personal relationship he felt with the Queen of Heaven?
I sense that Tolkien was disturbed and perplexed by the 'Protestant' direct approach to Jesus, which struck him as arrogant and over-familiar - and seemed to grate quite badly on Tolkien in the Christian works of C.S Lewis.
(Although it needs to be noted Lewis himself steered-clear of many subjects he felt were 'above' him - such as the nature of differences between Christian communions and the re-unification of Christendom.)
In sum, Tolkien's view of the nature of the world was profoundly hierarchical, non-symmetrical and incremental. It was therefore each person's primary duty to reverence that which was above him; and to accept that most people, most of the time, could neither understand nor relate to the ultimate nature of reality.