Serious question: where would one seek shelter if that notification were true?
If that ain't an option, jumping into the ocean doesn't seem like such a terrible idea.
For that - take shelter inside, anywhere. Make sure you shut all windows and doors tight, and ideally tape shut vents. Cover openings with a wet towel. Stay put (don't go outside!) and shelter in an interior room for at least one and ideally 3-5 days. Radiation follows an exponential decay curve, so waiting longer significantly reduces the amount of radiation you are exposed to. You have about 45 minutes after the blast (~1 hour after launch) before fall-out starts arriving in significant quantities: enough to take shelter and do basic preparations, but not enough to travel any significant distance.
The biggest physiological danger comes from inhaled alpha-particle emitters, i.e. radioactive dust. Alpha particles have only a few cm range and are easily stopped by wood, paper, or skin, though, so don't breathe the dust. Beta particles are also easily stopped by clothing or a few meters of air. There's not much you can do about gamma rays or X-rays (this is what your hypothetical lead-lined vault is for), but they are physiologically less dangerous than alpha and beta particles.
>Salt in seawater readily absorbs neutrons into both the sodium-23 and chlorine-35 atoms, which change to radioactive isotopes. Sodium-24 has a half life of about 15 hours, while that of chlorine-36 (which has a lower absorption cross-section) is 300,000 years; the sodium is therefore the most dangerous contaminant since it has the shortsest half life.
That’s a terrible idea. You’d be directly exposed when at the surface and end up inhaling fall-out.