LOTR has a religion in a sense that the characters sometimes sing old songs that hearten them in times of peril. They don't attend services weekly... they don't judge each other about religious views. The dwarves revere "Caves" while the elves wildernesses.
The good characters behave good, the bad characters bad. They play their roles.
The characters have a spiritual rather than religious nature. There's a creation myth involving Illuviatar in "The Silmarillion." But this brings in Mythology.