Short answers, pulled from the story.
Tolkien described The Lord of the Rings in a letter to his friend Robert Murray as "a fundamentally religious and Catholic work; unconsciously so at first, but consciously in the revision." He also said directly: "Of course God is in The Lord of the Rings. The period was pre-Christian, but it was a monotheistic world."
Tolkien wrote that explicit Christian teaching in fantasy would be "fatal," draining the life from a secondary world. He removed "practically all references to anything like 'religion', to cults or practices, in the imaginary world" so that the religious element would be absorbed into the story and the symbolism rather than stated directly.
Scholars Peter Kreeft and Jean Chausse identified Gandalf, Frodo, and Aragorn as embodying the Old Testament threefold Messianic symbolism of prophet, priest, and king respectively. Gandalf dies and returns as a transfigured figure in white; Frodo carries a burden of evil to its destruction as Jesus walked to Golgotha; Aragorn heals those stricken by the Black Breath of the Nazgul using the herb Athelas.
In a 1958 letter to Forrest J. Ackerman, Tolkien wrote that lembas carries "a much larger significance, of what one might hesitatingly call a 'religious' kind." Christian commentators read lembas as Eucharistic symbolism: the word "waybread" translates to viaticum, the Eucharistic food for a final journey, and in The Silmarillion the lembas parallels God's gift of Manna to the Israelites during the Exodus, described at Exodus 14.
Tolkien scholar Tom Shippey noted that Tolkien deliberately chose the 25th of March for the destruction of the One Ring because it is the traditional Anglo-Saxon date of the crucifixion of Christ and of the Annunciation. The quest also began in Rivendell on the 25th of December, the date of Christmas, bookending the story with dates from the Christian calendar.
Theologian Fleming Rutledge argued that Tolkien's solution is structural: the divine will operates almost entirely beneath the surface, signalled through statements in the passive voice. Gandalf says Bilbo and Frodo were "meant" to have the One Ring while leaving them free to act on that purpose. The Valar, Middle-earth's representatives of divine providence, are named directly only once in the entire text of The Lord of the Rings.