Nuclear weapon while stable (stored) ?

How much if any does a Nuclear weapon which is stored gives off radiation?

I'm curious because MUTO was looking for source of radiation and he went for a Nuclear weapon, right? But does nuclear weapon, while stored, gives off radiation that much?

I would assume it does not, because they would not (I think) leak radiation if the protective shells are made of quality materials which often are heavy chemicals elements. The heavier are materials = more radiation protection that's why lead is good cheap solution if you want to protect something from heavy radiation like gamma radiation. Alpha radiation you can stop with one sheet of A4 paper and beta radiation with aluminium foil.

We even see the army men interacting with the Nuclear weapon without any protection against radiation (if there was) from Nuclear weapon.

After all, Nuclear weapons are not supposed to react/decay rapidly when not "in use". If they did, they would be unsafe to store for a long time, and would deteriorate.

Lets assume it gives off just a little bit more than normal but not so high as to be hazardous for humans without any protection suits.

How is this enough radiation for MUTO to be attracted to and even lure him with it to the open ocean? (the military plan in the film)
That nuke is almost not even radiating.