Mitsubishi would be able to do it in the same shipyards as their largest cargo carriers and cruise ships but that might trigger an international incident. Maybe have pieces built in many different countries and brought together for a final construction elsewhere. Guns will be tricky to do for a variety of reasons, and the all-metal construction will cause a lot of additional problems. Updating the ship to modern tech will help somewhat, but engines will remain a problem regardless (civilians using nuclear ship engines will have trouble obtaining repairs if needed and many countries might not even let you near port). China could build the ship itself as could France, Germany, UK, and the US. Why a warship to exact specifications?