What Mark has stated is correct as with many radioactive isotopes Radium was discovered by Madame Marie Curie.
It has a half life of over 1000 years, this is the time it takes to an isotope to decay until half its original strength.
The dial painters died probably due to ingestion of radium.
I wouldn't worry about wearing a Radium painted watch the majority of Alpha & Beta particles can be stopped by a single sheet of paper, the fact it is behind on one side glass and on the other a movement, stainless steel case the possibility of ionising human tissue would be low.

Never got an A level in any science but I was an industrial Radiographer using Radioactive isotopes daily with no adverse effects....the baldness is natural...lol