Almost certainly, there is no such right.
It's illegal under 18 USC 831 to possess "nuclear material" without specific authorization. 18 USC 832 forbids the possession of a "radiological weapon". If there is intent to use the device to cause death, serious bodily injury, or damage to property or the environment, that's also a violation of 18 USC 2332i.
I don't think these laws have been explicitly tested against the Second Amendment, but related cases suggest they would hold up (if the challenge wasn't simply dismissed as frivolous).
The Second Amendment doesn't grant a blanket right to own weapons. Federal law, 18 USC 922 (o) makes it unlawful to own a "machinegun" (as defined in the statute), and in the case of Hollis v. Lynch, the Fifth Circuit held that this law was constitutional, because, as they said, the Second Amendment only protects weapons that are in "common use [...] for lawful purposes like self-defense." This case doesn't seem to have been appealed further, but the reasoning cited by the Fifth Circuit comes from the Supreme Court's opinion in D.C. v. Heller.
If machineguns aren't in "common use", and therefore not protected, surely the same would apply to nuclear weapons.