Earlier quoted context omitted.
There was an airburst test that battleships survived, but the nuke was way off target, even well shielded animals onboard died so if it was crewed it would have been a floating coffin and the nuke was a 20 kiloton device. One single Trident missile holds up to twelve 500 kiloton bombs.
1940s battleships were designed for just enough shielding from chemical bombs and bullets. We can assume that future battleships will have comprehensive radiation shielding. Of course, building a fleet to soften up the Chinese coast is more an exercise in science fiction than military planning.
We would do better to assume that future battleships won't get built, and that future surface combatants for the foreseeable future will be all carriers and destroyers.