I really doubt that it's radium. The clock does not appear to be old enough to have been made during the early radium craze where no markings were required. In later years (Typically Post WWII), they were required to put some type of symbol down by the 6 on the watch/clock face. (Not neccessarily denoting radium, but denoting that it contained a radioactive paint).
In the later part of the era in which self-illuminating paints were used, Promethium-147 was used because it was less problematic than the radium paints were and depending on where the manufacturer was located it was cheaper than radium paints as well. (China and Japan made great use of Promethium based luminescence, and the indication of 'MADE IN CHINA' on your clockface makes me think that perhaps it is a Pm-147 based paint. Again, however, you would expect to see some indication of the promethium paint by some sort of label).
If the clockface used Pm-147, you'll have nothing left at this point in time. The half life of a couple of years means that based upon a stop date of 1976 for the use of radioactive paints (I know they stopped in the 70's, but I can't recall the last year), your Pm-147 would have gone through about 15 half-lives. That means you have 1/2^15 of the original tiny amount of promethium left.
While the phosphors do die over time, they don't die completely. If you take your clock and keep it in a dark place for a day, then look at the face in the dark, if it used a self-illuminating paint you should see the tiny faintest of glow from the painted areas. It may not be much, but if you match it to a non-radioactive based paint you'll notice it. If there is absolutely no glow, then there is no activity left.
Honestly, the best way to tell is to pick up a geiger counter, or an even cheaper method is to pick up some polaroid film. Take one sheet of the polaroid film and put it right down on top of the watch face and leave it there for about 24-48 hours. Then open up the film to develop it. If the face is using radium, the intense gamma rays will expose the covered film in the area that the paint exists and that will show up on the polaroid. This is the easiest, and cheapest if you don't have a geiger counter, to see if Ra paint was used.