Omega and...radio on the dials

Yes, the dial most likely has radium in the 3/9/12 plots, as well as in the hands and the ends of the other hour markers.

I suspect that the dial may have been repainted so the luminous compound could be very early tritium, but only a radiation detector will tell for sure.

If I remember correctly, Omega stopped using radium lume on hands and dial markers in the early 1960s, maybe 1962 or so. OP, if you're concerned about any potential danger to your health from a radium dialed watch, I think that the general consensus is that the risk is low provided that you don't open the case and inhale any dust that might be present from disintegrating lume. It's also a good idea to store watches using radium in a well ventilated area because radon gas is a natural byproduct of radium as it decays.