From what I know very few countries are authorized to possess nuclear weapons, although a few others might secretly have some. But using nuclear power for civilian use (to produce electricity) is allowed for mostly any country, as long as they proceed with care.

But what about nuclear-propelled military icebreakers/ships/vessels/aircraft carriers that do not hold nuclear bombs?
Those aren't weapons of mass destruction, but do they fall under military nuclear power?
The nuclear reactions going on in those engines are perfectly controlled, and are mostly the same as those used in civil reactors.

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    The same question could apply to nuclear medicine, which uses nuclear technology for diagnostic or therapeutic purposes. I would expect all modern military hospitals to use X-rays, PET etc. May 29, 2019 at 8:48
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    Have any non-nuclear-weapon states ever operated military vessels with nuclear propulsion? I know that Japan and West Germany built civilian ships with nuclear propulsion at one point, despite being non-nuclear-weapons states; but I don't know about the military. May 29, 2019 at 14:52
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    You seem to think that the military operating a nuclear device means that the nuclear device is a nuclear weapon. That is not the case. May 29, 2019 at 15:00
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    @MichaelSeifert I wrote the question thinking that only nuclear-weapon states had nuclear propulsion on military vessels. cpast gave a really good answer, and after a google search, it seems that Brazil does have one of those now May 29, 2019 at 15:00
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    No country is "authorized" to possess nuclear weapons; they take it upon themselves to do so. Indeed, rather than being authorized they are obligated to eliminate them: The nuclear-weapon signatories to the NPT committed themselves "to undertake effective measures in the direction of nuclear disarmament". May 30, 2019 at 0:38

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The treaty you're thinking of is the Treaty on the Non-Proliferation of Nuclear Weapons (also known as the NPT or NNPT). The core of this treaty for non-nuclear weapon states is Article II, which says

Each non-nuclear-weapon State Party to the Treaty undertakes not to receive the transfer from any transferor whatsoever of nuclear weapons or other nuclear explosive devices or of control over such weapons or explosive devices directly, or indirectly; not to manufacture or otherwise acquire nuclear weapons or other nuclear explosive devices; and not to seek or receive any assistance in the manufacture of nuclear weapons or other nuclear explosive devices.

As you can see, the limit is on "nuclear weapons or other nuclear explosive devices," not on all military applications of nuclear power. Another provision of the NPT reinforces this. Article III requires non-nuclear weapon states to enter into safeguards agreements with the IAEA to ensure that peaceful applications of nuclear energy aren't diverted to make nuclear weapons. In the guide to these agreements (IAEA INFCIRC/153), paragraph 14 covers use of nuclear material for military purposes. Countries are generally expected to notify the IAEA if they're doing this and give assurances that they won't use the material for weapons, but other than that it's perfectly fine to use the material for non-peaceful purposes. The country does not have to tell the IAEA any classified details about the use or get their approval for the military activity, just that X amount of Y material is being used for military purposes. Countries have used this provision when considering naval propulsion in the past.

Countries might also take the position that naval nuclear propulsion is a peaceful use (which would make them subject to safeguards, but might give advantages under other treaties). For instance, Brazil and Argentina entered into an agreement (INFCIRC/395) to exclusively use nuclear power for peaceful purposes, but in Article 3 declared that nuclear propulsion is a peaceful use. In their IAEA safeguards agreement (INFCIRC/435), Article 13 essentially restates paragraph 14 of INFCIRC/153 but changes some terms to be consistent with that position. "Material not subject to safeguards" becomes "material subject to special procedures" and "non-peaceful applications" becomes "nuclear propulsion," but it's otherwise basically the same. Whether or not nuclear propulsion is a peaceful use, this agreement (which the IAEA signed) shows that it's fully compatible with the NPT.

Because naval propulsion is exempt from detailed monitoring under safeguards agreements (all military use is, but naval propulsion is basically the only practical military use for nuclear energy other than weapons), it's been brought up as a potential loophole in the NPT. Naval propulsion also tends to involve much more highly-enriched uranium than civil power, so the concern is that a country could divert highly-enriched uranium from naval propulsion and use it to build weapons. However, the current rule is that naval propulsion is absolutely an acceptable use for nuclear power even by a non-nuclear weapon state.

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