Tolkien's work is unique in many ways. There is a fairy tale quality to it, and everything that came before it, such as William Morris's The Well at the World's End or Eddison's The Worm Ouroboros. The former is a stylized version of medieval lays, and the latter is an extremely baroque reframing of Norse legend. In fairy tales and mythological works generally, there is often much faith and little religion. Looking at Germanic sagas, there is often not even a mention of religion, because the world is one where the gods are immanent (I mentioned this earlier in my replies) in the world.

We see this in Tolkien, given that we have several divine (or fallen) beings in the Hobbit and Lord of the Rings: Gandalf, Saruman and Radagast are divine beings sent from the West, the divine realm that is mentioned at several points in the trilogy. The Balrog of Morgoth in the mines of Moria is a demon. Sauron is clearly a Satanic figure (though we learn that he is secondary to Morgoth by mention, if not as thoroughly as we do in the Silmarillion, obviously).

There are also the numerous professions of faith which are, yes, usually in Elven ( A Elbereth Gilthoniel silivren penna miriel... ), but mentions are made of the Star of Earendil (a corruption of the Anglo-Saxon name for the morning star; this is of course somewhat ironic given its Luciferan connections) and the light shining in the darkness, etc. There are hints of religion even, though the story by its nature shies away from them.

Because Tolkien was the one to popularize the fantasy genre, and because he drew on fairy tales for inspiration, and because the entire trilogy was a metaphor for his faith, I think this "light touch" of religion is understandable. Even so, there is a faith there, as we find out when we read other books.