[COLOR=Red]Certainly not - unless your watch was made no later than 1930.[/COLOR]

Not so. I've got a watch made in the 70's, itís one of those self-winders with a luminous dial. It's the luminous dial that gives off radiation.
Many if not most of the workers in the UK who assembled instrumentation on ww2 fighters and bombers (a luminous paint was used on the dials) got throat cancer. They used to find it helpful to lick the brush now and again I believe. I'd imagine the pilots got a fair old dose considering the number of dials and how crude the luminous paint was, but that was the least of their worries!

A few years ago one of the mechanics at work (I work in a nuclear power station) set of the personnel monitor on leaving the nuclear island. It turned out he had his good watch on that day by mistake, it was an Adidas model (I think)with a luminous dial. It was a modern watch anyway. Strictly speaking it should have been declared a source and not allowed off site. As the contamination was 'fixed' he was allowed to take it home. He was also advised to contact the manufacturer for a replacement or refund, as while it was a weak source it was clearly not a good idea to strap it to one's wrist for days on end.

Like wise that smoke alarm. Stick it in your pocket and given some time it will have a detrimental effect on your health. However screw it to the ceiling and it's perfectly safe. As we say at work, double your distance, quarter the dose.

Oh I still have my old watch, but I certainly don't wear it anymore.
As for radioactive lenses. I'd expect the radiation to be much lower than that of luminous watches and fire alarms etc, but I'm only taking a guess.

Your local school, college, university, hospital or nuclear power staion should be able to check and advise.