> I don't see how sunk steel would be any more valuable than steel from freshly mined ore. But I like to be surprised about these things :-)

Steel production uses air from the atmosphere. Thus it picks up the increased background radiation while it is refined. It may be possible to scrub the radioactive components from the air to avoid contanimating the steel, but I expect the cost would be prohibitive (at the very least, more expensive than getting it from old battleships).