What is the radiation danger from nuclear submarines that are sunk?

Very small. The radiation is shielded by all the water that is around it. As a rule of thumb... for every 2 feet of water, the amount of radiation is reduced to about 10%. (so for 4 feet of water it's reduced to 1% of it's original value) As long as the core is intact and surrounded by water, almost none. There are a wide range of variables, like what happened when the boat went down. Did it implode? Crash into the bottom at speed? Usually, even if implosion and a crash occur, the danger isn't great. Far and away the greatest radiation hazard is from damaged and leaking fuel elements. They'll put nasty stuff into the water, but they're pretty safe welded into a core assembly and stuck inside the pressure vessel.