The short answer is, don't worry about it, but don't eat the dials either.
Radium was used after about 1900 as a means of illuminating watch hands and markers at night. It was widely used until the 1940's or so, when the hazards of radiation were (belatedly) understood. Since them, a less powerful radioactive source, tritium (a kind of hydrogen) has been used. At the time of this writing (November, 2000), tritium is being phased out of watch dials, partially due to availability of tritium, and partially due to the development of non-radioactive luminous compounds like Luminova which "hold their charge" of light better.
The main victims of radium were the watch dial painters, who were encouraged to keep a fine point on their paint brushes by licking the brush end. The wearers of the watch receive only a small dose of additional radiation per year, much less than the natural background radiation.
A thorough examination of the issue is contained in Bruce Lulu's definitive article on luminous watch hands.

7.1 What's the difference between acrylic, mineral, and sapphire crystals?

An acrylic watch crystal (i.e. the see-through window above the dial) is a kind of plastic, which has the advantage of being resistant to shattering, but can be scratched easily. Fortunately, scratches can be easily removed from acrylic.
Mineral crystal is a kind of glass, which is more scratch resistant than acrylic, but not as as good as sapphire. Mineral crystals are also somewhat better at shatter resistance compared with sapphire. Unlike acrylic, scratches in mineral glass are more difficult to buff out; and unlike sapphire, mineral glass will scratch. In my opinion, the mineral crystal seems to be a poor compromise between the two extremes.
A sapphire crystal is indeed made of synthetic sapphire, which is a transparent form of corundum, or aluminum oxide (Al2O3). It is extremely hard (Moh's scale 9), and will resist scratching by most substances short of diamonds. However, if struck sharply and from the correct direction, sapphire will shatter. Despite the relatively large size compared with sapphire gemstones, sapphire crystals aren't very expensive (~$20). (Another fun fact - the windows built in to grocery store checkout lines which scan for the bar code on products are often made of synthetic sapphire - for the same reasons that watch crystals are!)
7.2 How do I remove the scratches from an acrylic crystal?

Good old-fashioned toothpaste is a good start. Start by buffing a little bit onto a clean cloth over the area of the scratch. Rub lightly, and rinse with a very slightly damp cloth. Repeat until gone.
Note that there are also purpose-made polishes for this purpose, such as Polywatch or Crystal-Kleer
(Legalese: I cannot be held responsible if you try this for yourself, and end up ruining your watch. This procedure is intended for people willing to take responsibility for their own actions only.)