The nuclear warhead one is the most interesting case, of course. I have many times pondered this exact problem. It's a true analog -- maybe the person just wants the warhead to ensure property safety, or perhaps for research or other peaceful purposes. Like the original poster, I see no "natural" reason why we should restrict ownership of items that only pose a threat in criminal hands (such as a gun or a nuclear warhead). However, I would also agree that removing restrictions on the ownership of nuclear warheads is a pretty darn terrible idea.
In attempting to determine what exactly gives us the ethical right to restrict nuclear warhead ownership, I would have to argue that the main concern is public safety, with connected problems of international relations, global stability, and risk of permanent habitat damage. The problem, for me, comes down to this: can we handle this as a public safety issue on a case-by-case basis? With guns, I might argue that the answer is yes (though we're not doing a good job of it now). It's possible to effectively deter and punish gun crime. The results of errors in this deterrence or punishment, while regrettable, do not cause a major disaster of public safety, international relations, global stability, and permanent habitat damage.
Nuclear warhead explosions, on the other hand, are not subject to effective deterrents or punishments. One must be extremely motivated to use a nuclear warhead. Additionally, errors are inherently major disasters. While a dozen errors in gun deterrence/punishment might lead to a dozen deaths, a dozen errors in nuclear warhead deterrence/punishment would potentially lead to global nuclear winter.
No matter what, we run into a sticky problem: how do we balance individual rights with the need for public safety? Statistically and scientifically, gun bans have been shown to increase overall public safety. Is that enough to overcome what I would consider a "natural" right to own whatever we like? I don't know, and I don't think there is an easy answer.
That said, we're ignoring our own constitution with current gun laws. Personally, I find that reprehensible. If we want to count that as a part of our law, we should adhere to it. If we want it to say something else, we should change it. We have the tools for that; it's called the constitutional amendment. Pretending that it says what we want is a much worse problem.