- The construction cost would be significantly higher than a conventional ship.

- Reactors are far from trivial, so you'd double or triple the crew required.

- Shipbreaking would become even more of an issue than it already is. You can't just beach a ship like this in Bangladesh and have a bunch of untrained people attack it with plasma cutters.

- The ship would be a huge target for pirates and terrorists. It's essentially a floating dirty bomb, after all, just waiting for the USS Cole treatment.

- A lot of countries would not accept nuclear ships, both due to perceived security risks and for more ideological reasons.

... and that's probably only the tip of the iceberg.

Nuclear is barely economically viable with land-based large-scale nuclear power plants running for 50+ years. They are an attractive option for some military ships, but I doubt anyone would be willing to risk it for regular commercial shipping.