Several reasons that more or less interlink with one another, I think.
First, what kind of religion could that have been? Going with the conceit that Middle-earth is our own world in some imaginary age out of the past (B.C.), it would have been unbelievable for its people to follow the Christian religion their author held with, or anything closely similar; if, on the other hand, we consider Middle-earth as a self-contained sub-created world, any inclusion of or allusion to real world religion would have been detrimental to its autonomy - or in simpler worlds, would have broken the spell.
On the third hand
, I imagine that Tolkien - exactly because he was a devout Christian himself - somehow didn't feel it within his rights as a subcreator to 'make up' a religion for them; maybe he also felt it would lessen the dignity of his characters if he had them holding a plausibly pre-Christian 'pagan' belief that would have been contrary to what he himself held to be true. So in the
Silmarillion, he walks the line by having the Valar acting like the Gods of the Norse or Greek pantheon to satisfy his mythopoetic desire, but making them not true Gods but angelic powers under Ilúvatar, thus appeasing his religious conscience; while in LotR he does his best to avoid the whole issue altogether by making no overt mention of his characters' religious beliefs and customs at all but rather absorbing the religious element into the story and the symbolism (as he put it himself in Letter 142).
This decision, of course, has the benefit of allowing him to present the truth he believed in a way that appeals to readers of widely different cultural backgrounds, whatever their own religious or philosophical convictions.
For previous discussion of the matter, see
this thread and
that one.
(x-ed with everybody else)