Tolkien's work has pretty much been a constant in my life ever since I began pondering more metaphysical matters - until then I just went along with what was on offer, which was going to Sunday School and listening to the people around me. After I found Tolkien was about the time I started to make my own mind up, something I was more or less encouraged to do anyway.
I had C of E parents, a mother who does believe in God, a father who wants to but says that really he can't - he seems to like churchgoing more for the social aspect, haranguing vicars about the existence of aliens, and some kind of sense that the steady, conservative (with a very small C, as his politics are anything but big C) England of the 1950s has not entirely gone away.
His father was CofE but never went to church, his own creed being more political and left wing; his mother was brought up Catholic, and after marrying my grandfather gave it all up publicly but never really did in her heart - my childhood was somewhat afflicted with her visions of a cruel God who might smite me at any moment. My father once 'had words' about her allowing me to read a Catholic catechism filled with concepts I did not understand. My mother had a CofE father and a mother who was disowned by a catholic family. I also had Methodist (one a Welsh great-great grandfather who was a Methodist preacher who refused to learn English) and Jehovahs Witness relatives. My RS teacher was a Quaker who said "Jesus was a Communist" and taught us to challenge our own pre-conceptions. I've also learned Tarot from a woman who learned it from her mother (and so on backwards), have studied astrology and wicca and have learned about shamanic practices. So really I've no reason to favour any religion over another.
Like a lot of people, not long after reading Tolkien my interest was stirred in mythology, I developed a keen sense of the environment around me and then coupled with the left-leaning politics I had also developed an interest in, I rejected my own religion. I do have to say that Tolkien was a pretty big catalyst in my becoming an atheist, and what he would have to say about that I do not know. I saw a keen moral sense of right and wrong in his work, but I did not see that it was anything to do with God.
After many years I have explored all kinds of ideas, and have decided that I'd rather there be a world with God than without, but I do not follow set creeds as they aren't for me. I suppose my strongest influences are Socialism, Paganism and Christianity. What I see in Tolkien's work is a world where there is God, and where there are right and wrong things to do, but which is not governed by churches or specific religions and their associated divides. It's a world where the individual must think for themselves about what is right and wrong, and where even the seemingly terrible person can be forgiven. Strangely, given Tolkien's Catholicism and his (small c) conservative leanings (yet spiced with a hint of anarchism) I find his world to be presented in a not dissimilar way to how I see it.
I think given that we have so many readers here who are from such different faith (and otherwise) backgrounds and yet who all respond to Tolkien's work so deeply, his own personal faith cannot be written in big bold letters across his work. If it was, then wouldn't some of us be repulsed by what we read? I think Tolkien tapped into something far greater than individual religions.
Are there specific things which people of certain beliefs find they can't agree with in Tolkien? If there are, then maybe his work is not all that universal, but as it happens I think our personal religious or otherwise beliefs are not that relevant in whether we enjoy his work or not.