Entirely possible that radium was used for longer in the dash timers. Firstly, on health grounds, as no-one was walking around with a dash timer strapped to their wrist (although I am sure that there were a few more eccentric individuals who did!) and secondly because the luminescence is brighter for something like a rally night stage or a 24-hour race.

Tritium and radium lumes are both self-luminescent, that is they glow all by themselves without having to be exposed to light like the current photo-luminescent materials we are used to from the end of the 60s/start of the 70s til the present day. The process is based on radioactive decay illuminating various phosphorus compounds - the make up of the phosphor determining the final colour of the lume. This decay happens equally at night as by day, so was ideal for night-time applications like the racing above and aviation etc. The reason radium gave a better glow than tritium was simply that it was more strongly radioactive, thus bombarding the phosphors more often. However, while the beta particles given off by tritium are blocked by human skin (and would pretty much always have been absorbed into the watch before ever getting there), radium gives off more penetrative and also more dangerous alpha particles. Science uses a horribly bland euphemism for possible cell damage from radiation, "relative biological effectiveness" and that is much higher with alpha radiation than with beta or gamma - the chance of cancers and/or cell death is much higher with alpha and thus radium than the beta particles from tritium. The range of a beta particles is a few millimetres, an alpha particle a few centimetres and this obviously has an impact too. Though skin is a fairly effective barrier for most alpha radiation too.

Radium gained widespread usage through its early discovery (yep, Marie Curie as I'm sure you all remember from science lessons, however far back they were) and the strength of its radioactivity. Its half-life also gives it the upper hand over tritium for use in lume materials - the half-life of tritium is just over 12 years so it loses 50% of its effectiveness in lume after that time, whereas radium-226 has a half-life of just over 1600 years and will only lose 1% efficacy in 25 years! Other factors are in play, so the phosphor elements in lume can absorb water and break down, so the lume in an older watch or dash timer may no longer work, but if you take a Geiger counter to them, you may well find that the radium is doing its part of the job quite happily still, even if there is nothing left for it to make glow...

Phew, it's a long time since those Physics and Chemistry A-Levels but they still come in handy sometimes! As does Wikipedia :)