I built a timing and lap counting setup for a slot car club a few years ago. I used IR LEDs in an overhead gantry, and IR sensors fitted into holes in the track. I also built start countdown lights (F1 style). The sensors and lights were run from the parallel port of an old laptop, and the software displayed lap times and race positions in real time. Unfortunately I'm not a member of the club anymore, and I can't remember the name of the software. I think it was either free or very cheap.
Before settling on the above/below IR method, I tried visible light (too much interference from ambient lighting), bouncing IR off a strip of metal tape under the cars, and reed switches to detect magnets on the cars. IR was pretty much perfect, although we had to switch off the high-level radiant heaters in the hall during each race.
A bit harder to do for a small RC track like yours, as the cars don't stick to individual lanes, but I think you only want to time one car at a time?