As suggested in a sister comment, a nuclear reactor is such a highly sought piece of tech that you need an aircraft carrier built around it to protect it from snatching by somebody who wants to start building nukes.
You can, of course, use nuclear energy to run your ships, and even trucks and planes.
Use the nuclear energy (both the electricity and heat) to produce fuel from CO₂ and H₂O. Use this fuel in your jet engines and colossal diesel motors. Burning it is carbon-neutral.
Oil should become much more expensive though, and building of nuclear facilities, much more cost-efficient, for this to become economically viable.
State != military, though. For instance, all France nuclear power plants are operated by Électricité de France (EDF) that is substantially owned by the French Government, with around 85% shares in government hands ([1]).
I don't see a problem having nuclear container ships to be operated by civilian governmental agencies / state-owned enterprises. This might not work for the US, but we often have very unique ways of doing things.
i suppose these ships only use Soviet / Russian ports, and mostly don't operate in high seas. Icebreakers do operate in high seas, but getting close to them is a bit hard due to ice and polar weather.
The more interesting example is NS Savannah [1], which indeed crossed oceans unattended, and was accepted at European ports.
Of those, the icebreakers have been commercially successful due to logistics of fueling in the attic and the heavy fuel needs of ice breaking (both solved by a reactor that only needs to be refueled once every few years).
You can, of course, use nuclear energy to run your ships, and even trucks and planes.
Use the nuclear energy (both the electricity and heat) to produce fuel from CO₂ and H₂O. Use this fuel in your jet engines and colossal diesel motors. Burning it is carbon-neutral.
Oil should become much more expensive though, and building of nuclear facilities, much more cost-efficient, for this to become economically viable.