The cell phone radiation myth has nothing to do with the pre-1960 watch lumen problems. Cell phones only emit radio waves in the VHF band at relatively low energy, so please do not confuse them with watch lumen.
Radium 226 isotope is not found in cell phones as far as I am aware but was mixed in the lumen paste of many pre-1960's watches, sometimes quite liberally in "military" watches and clocks. Radium 226 has a half-life of 1,600 years whereas the paste material that glows from the alpha and gamma radiation emitted from R226 only lasts for 20 years or so because it "burns out". Thus the lumen of many old watches and clocks has ceased to glow, even when placed in bright light. Nevertheless the R226 still emits gamma, which creates the dangerous by-product radon gas.
A quick test of old lumen. If the old watch lumen glows brightly after exposure to bright light and then fades after a short while, the lumen is unlikely to have radium 226 as the exciter. If the old lumen does not glow or glows extremely faintly after being exposed to bright light, then that lumen is likely to have radium 226 as the exciter, still emitting gamma and alpha radiation. Alpha will not penetrate the watch case/crystal or human skin but gamma is extremely penetrative (through at least 6 inches of lead).
Old radium 226-excited luminous paint is a serious problem, not just for old watches. It is found painted extensively on old classic aeroplanes' and old ships' instrument dials and switches. The engine RPM gauge and airspeed indicator of the 1951 Chipmunk I fly emits gamma radiation in excess of 10 µSv/hour (off the scale of my hand Geiger reader). As the endurance of a Chipmunk does not exceed 2 hours per flight, I live with the emissions, as I had done for several years in the 1960's military.
Radium 226-excited lumen dust from inside a watch is deadly if inhaled or ingested because the emitted alpha will attack human organs, once inside the skin. Radium 226 will find its way via the blood stream into bone structure. Thus "not licking the dial" is excellent advice here!! Our watch repairers generally have the problem and my guy is very aware.
Best
Clive