This question is much more in depth than it may appear; thus giving an answer is much more difficult than may be expected.
Libertarians (and anarchs) generally believe that no groups of people have any more right than any individual. For instance, I do not have the right to steal your car. Thus, no group – regardless of size – has a right to steal your car; though groups known as government claim such an illegitimate right. In a similar fashion, every individual has an inherent right to defend themselves (including their friends, family and others from imminent attack) in any manner they see fit. That includes the right to own any weapon they believe is necessary to defend themselves.
Whether or not this right includes the right to own a nuclear weapon has been the subject of many debates. I believe that, as stated above, no group has more rights than any individual. Therefore, if any group claims the right to own a nuclear weapon, every individual has the same right to own said weapon. Though, I do not believe that any person or group (including any government) could legitimately use a nuclear weapon in a manner purely consistent with self-defense. Any discharge of a nuclear weapon would, no doubt, kill or injure someone not involved in any attack leading up to the supposed self-defense discharge of said weapon.
I realize this doesn’t exactly answer the both questions, the subject needs a bit more explanation and historical perspective. Before the “Civil War” began in 1861, James Buchanan was criticized for not doing anything to prevent the southern States from seceding. Buchanan believed that while no State had a right to secede from the Union, the federal government had no right to prevent the State(s) from doing so. I take a similar approach in regards to nuclear weapons. While I believe that every person and group of persons have a right to own any weapon they believe is necessary to defend themselves, and I do not believe that any nuclear weapon can be used in legitimate self defense, I also do not believe that any person or group has the right to prevent anyone else from creating, building or owning such weapons unless the person or group attempts to use the weapon.
One final question that I will not answer: since the U.S. is the only country to ever use a nuclear weapon in war; why should that government be responsible for deciding who gets to own such a weapon?