I am in no case planning on disassembling the watch and compasses that I have but I have kept them in my bedroom where most of my collection has been stored for quite some time now. The Elgin will glow in the dark at night. The paint on the arrow in my Taylor compass broke and fell into the liquid. As Doyler says there are many other things in life that are equally or potentially far more damaging than a small strip of radioactive paint, but perhaps it would be in the best interest of collectors to provide this community with some sound advise? Obviously watches and compasses which are covered in glass will be far less hazardous than say a German field police gorget which has this same type of paint exposed on the surface.
I've done a bit more research on this topic and have found that there are no less than 3 types of radiation this paint will emit: alfa radiation, beta radiation and gamma radiation. Gamma radiation seems to be the one most of us should concern ourselves with as it will travel through matter, including metal and glass casings, sort of like an X-ray. Alfa and beta radiation appear to not pose any risks as long as the watch or other instrument remains intact and unopened. Unless cracked, the casing will stop these types. As you have stated, General Apathy, through the experience of the Dutch repairman, although less effective in traveling through matter, alfa and beta radiation are the real hazard, exposing yourself to that stuff without any protection seems to be a very bad idea. That doesn't mean gamma radiation should be taken lightly. Sure you can't control every unsafe thing in life, but I reckon there are preventive measures one can take, a difference that can be made, to lessen the likelihood of becoming sick because of these items, something along the lines of not wearing these watches or not having one on your nightstand?
So, what's the safe, middle ground here? I'm sure boxing one or two of these watches and burying them in concrete for the sake of safety could be a bridge too far, I mean, we are still collectors. While browsing though I've come across huge collections of WW2 compasses and watches, even on here, which could raise some safety concerns for the collectors who, say, got a tad too enthusiastic.
I am guessing that in any case for most collectors it would better to remove the few compasses and watches they may have out of the bedroom, if that is where they are kept? Disposal perhaps would be an overreaction, many of us are attached to this stuff and there will always be things that are unsafe, from the air inhaled to the food consumed, but where would you draw the line? How many watches or compasses are the limit to store in 1 household/collection and where would be the best place to store them: attic, cellar, anything but your bedroom?