'...consciously in its revision.'
Although Tolkien didn't originally set out to write Christian and/or Catholic fiction, his Catholic mindset naturally played a part in everything he wrote. Even in something as operatic as The Children of Húrin, there's a sense that the tragic deaths aren't the end of the story, that Húrin's being held in captivity by Morgoth isn't the only result that will come from the Man's faithfulness. Because of what we learn in the first chapter of The Silmarillion, we know that the discord sown throughout it by Morgoth and Sauron will be woven by Ilúvatar into a music that will be even more beautiful because of it. We don't see the last chapter or hear the final music, but that's very Catholic in itself. With everything the Church has lived through over the last 2000 years, it's not surprising that the only way she's found to honestly deal with "the problem of evil" within the will of a loving God is through mystery and admitting that we finite creatures can't understand how it will all play out. It's even understandable for Tolkien personally if we remember that he first began writing his mythos into prose as he lay in the hospital after all but one of his close friends had been killed - following a childhood of poverty and being orphaned at the age of 12. It's perhaps less surprising that he was drawn to the tragedy inherent in "Northernness" than it is that he was able to keep it from leading to despair.
But another thread began winding its way through his mythology, and I think it's entirely fitting that it came by way of the stories he invented for his children. Many Tolkien scholars consider Roverandom to be the beginning of his higher mythology breaking into his children's stories, but I think it could be as truly said that it's the beginning of the hope of his children's stories breaking into his higher mythology. What happens when you have a toy dog catch sight of the Undying Lands?
This integration is more explicit in The Hobbit. The stories about hobbits that Tolkien told his children had nothing to do with Middle-earth; hobbits didn't live in that subcreation. But by bringing Elrond into the story, Tolkien merges the world of hobbits and the cosmos of his mythology. That act brought other aspects of Middle-earth within the bounds of what was essentially a children's story, if a more mature one than Roverandom.
The Lord of the Rings began, long before it had that name, as a sequel to The Hobbit - as another children's story. There were a multitude of ways the story became much more than that as Tolkien rewrote and revised, and with every change that I can think of it became more Catholic. Gandalf shares what he learns about the Ring with Frodo, instead of hiding his knowledge from Bilbo and Frodo. Frodo leaves the Shire out of love for it - not because he's bored and has spent all his money. Gandalf returns after his battle with the balrog because he has passed through death and has been sent back to complete his task - not because the abyss wasn't really as deep as it looked. Boromir changes from a simplistic "bad guy" into a complex individual who struggles with real human frailties - and who finds redemption. The power of evil becomes more real as the Ring's effect on an individual goes far beyond making him invisible - which means that the power of good must also become deeper. Instead of killing Sharkey in a sword battle in front of Bag End, Frodo performs an exorcism, telling a "fallen angel" to leave and never return. Rather than being honored and happy on his return to the Shire, Frodo experiences another dark night that brings him closer to spiritual perfection.
In the process, hope is transformed into something deeper and higher and more powerful. The Lord of the Rings could never be subtitled "There and Back Again," as with The Hobbit. Its hope doesn't depend on returning to our starting place but on accepting the call to go beyond it. Hope changes, you might say, from an immature hope to a spiritually mature one. I believe that was possible because Tolkien was maturing as a Christian over the sixteen years that it took him to write it and, especially, to revise it - consciously - into a "fundamentally Catholic and religious work." The depth in which he thought about some of the aspects of the book is obvious in many of his letters.
When judging the Christianity of Tolkien's work as a whole, it's necessary to remember that, especially when dealing with the posthumously published work, much of it was written when he was a young man - at an age when C.S. Lewis was still an atheist. Tolkien was never an atheist, although he suffered grave doubts after two of his friends were killed in the same World War I battle that he was part of. Tolkien became Catholic at the age of four and he remained Catholic even though it made him something of an outsider in his own society. But the Catholic Christian he was in his teens isn't the same as the one he was in his twenties. The faith he had in his thirties isn't the same as the faith he had in his forties. His Christianity continually grew and matured and it's possible to see that reflected in his writing, if we match whatever it is that we're reading with the age he was when he wrote it (remembering that he kept - and Christopher published - much of the process along the way). I believe that, almost (although not quite) paradoxically, part of this maturing was allowing some childlike faith to break through into his adult writing.