Yes, the chapel was nondenominational. In Kirk's opening remarks at the wedding, he mentioned "our many beliefs." Seems to me that would imply that some of the crew had some sort of spiritual beliefs, maybe even religious beliefs.
Right! So obviously not everyone is an atheist.
That statement does seem to indicate not that people aren't
religious, but that people are free to believe or not believe what they like without fear of persecution. My reading is that in Kirk's time there are religions, both Earth based and those of other planets, and that people do believe in deitys etc... but just that the people who believe and the people who don't believe aren't in conflict with each other anymore. Everybody just respects everybody else's P-O-V.
Also in "The Final Frontier" it seems clear that Bones believes in God. Or at the very least he wants
to believe: "Jim, you don't ask the Almighty for his ID!"
Certainly by TNG, Roddenberry's ideals against religion (no doubt informed by the height of the televangelist craze of the 1980s) had become much stronger and definitely influenced the production. So 24th century humanity professes to not believe in religion at all, although I do imagine they would always be tolerant and accept it in purely scientific/historical terms (cf. "Haven" or even "Who Watches The Watchers", where the 1701-D crew have an appreciation for these cultures openly believing in Gods while still having no direct religious belief themselves). This is further enhanced in Deep Space Nine, where the Starfleet crew often appreciate Bajor having such strong faith while not actually worshipping at the altar of the Prophets themselves.