Read the section on page 24 of the DMG, "Bringing Back the Dead".
If you are playing in Faerun, then instead of traveling to an Outer Plane, you end up in the Fugue Plane. Since you don't have a divine patron, no one comes to get you. If you hang around you will eventually be judged by Kelemvor and stuck into the Wall of the Faithless. This might take years. This gives you a lot of time to think. You can voluntarily decide to hire on with the devils and go to Hell (yes, they recruit). Otherwise, you just wait until DM-Kelemvor decides you get judged and aren't able to be raised from the dead. Short of some shenanigans in the Fugue Plane, your soul is free during this time.
Even in Faerun, if someone tries to bring you back in that time period, the same rules from the DMG apply. You know the name, alignment, and patron deity (if any) of the caster, and can decide to come back or not.
By the book, it is absolutely irrelevent if your character believes in a deity. The DMG (and Forgotten Realms material for Faerun) describes pretty clearly what happens and what choices you have--belief or lack thereof doesn't enter into the equation of being raised.
End of that part.
Now, as to whether the deity of the cleric casting the spell will agree to bring you back, that is an entirely different aspect and moves heavily into personal setting design, DM style, etc. But as far as I can tell from the rules as written (and not just 5e, but correct me if 1e phrased it differently) once a cleric has a spell prepared/memorized, he has that spell in him, and he doesn't need his god to cast it.
After that is when the DM's decisions come in. Do gods actively step in to stop their clerics from casting certain spells that they already have prepared? Or do they just punish you for doing stuff against their ethos? When you prepare your spells, does your god have to grant them to you, or does being a cleric give you class abilities that are just as irrevocable as any other class? Some previous editions answered those questions (and not in the same way). 5e hasn't answered them, so it's up to the DM.
Personally, I think mythical religion provides some of the most interesting elements of D&D, and I like it to play a role in my games. I wouldn't feel a need to punish a character whose player wanted him/her to be atheist, though I would make sure the player understood how unusual it would be to be an atheist, and encourage them to have a good role-playing reason. Unless they are some sort of philosopher it would make a lot more sense to believe the gods exist but just not really care. Actively avoiding anything that we might consider religious would be pretty extreme, such D&D societies being saturated with religion everywhere, and quite possibly not making a clear line between "stuff having to do with religion" and "stuff not having to do with religion".
What I would do is allow the character to interact with their attitude or choice during the game. If they are actively atheist, it's something that will be constantly coming up. If they just don't care, they will see a world where a lot of good vs. evil and divine power happens, and can decide how their character deals with such situations. If the player is actively irritated by religion in the world, then they probably just wouldn't be a good fit for my table. I'd make sure they understand how that sort of stuff works before the game starts.