Could grandfather's war watch turn out to be poisonous?

I inherited a watch from my grandfather that was issued to him when he was serving in the war. The numbers and the dial are luminous, and I've read that this could mean they are emitting radon. Just how dangerous is this? I only wear the watch occasionally.

- Dr Rosemary Leonard

In the first half of the 20th century, paint containing radium was used in watches as this gave luminescence to the hands and dials. They were issued to servicemen as they could be useful for seeing the time in the dark.