There are bigger problems with nuclear powered cargo ships than just waste disposal. They would need larger crews of highly trained (more expensive) experts to operate the reactor. Look at how much the US Navy spends to train competent technicians and engineers.
Every nuclear powered ship becomes a prime terrorist target, either to hijack it or destroy it in place and cause a radiological incident. So you also need constant armed security (also expensive).
A more practical way to have "nuclear powered" ships would be to use electric power from land based nuclear plants to manufacture synthetic hydrocarbon liquid fuels, then load that fuel on the ships. Nuclear plants on land are far more cost effective since they can be built much larger to achieve economies of scale on operations and security.
On the subject of highly trained nuclear operators from the Navy, what's the job market look like for those guys after they leave the service? Where else are you going to find a nuclear-powered ship where you can put your experience to work?
Seems like a natural fit. For almost everything in the military, IMO there should be some market in the private space that could use those skills.
Some of them stay 30 years in the service and then retire. Others go to work for civilian nuclear power plants, or the vendors which service and supply them. And there aren't very many trained operators to begin with. The US Navy has <90 nuclear powered vessels (plus a few separate reactors for training and research).
Every nuclear powered ship becomes a prime terrorist target, either to hijack it or destroy it in place and cause a radiological incident. So you also need constant armed security (also expensive).
A more practical way to have "nuclear powered" ships would be to use electric power from land based nuclear plants to manufacture synthetic hydrocarbon liquid fuels, then load that fuel on the ships. Nuclear plants on land are far more cost effective since they can be built much larger to achieve economies of scale on operations and security.