I started producing nukes in a few different cities, and when making them, they simply state something like they add one more nuke to my reserves. Is there a limit to how many nukes I can have, or can I just keep stockpiling them?
I haven't seen any indication that there is a limit, but the maintenance costs are pretty steep at 14-16 gold per turn. So your stockpile is certainly limited by your wallet.
Nukes are not bound to individual submarines or silos, you just have a central stockpile and can fire from any of those. They don't have a capacity for nukes or something like that.
Nukes do have a limit, well no really you can have as much nukes as you want but, you need uranium. Even though you might have lots of uranium eventually the world well run out.