If there is enough of a variance in tire diameter, it could most definitely be the problem. If the front tires are registering one speed and the rear tires are registering another speed, the difference could be causing the ABS computer to have fits. The difference would need to be a bit though, but don't know exactly what the difference would be (is probably different when considering different ABS types/manufacturers).
While this might be the problem, there could be problems with the wheel sensors or the ABS unit itself. You'd be best served to get it to a mechanic who can read the ABS codes and see what is going on for sure (in other words: don't guess what's wrong, know what's wrong).
Personally, I would ensure this gets fixed prior to purchasing the car. If it's only the tires, have the owner get them changed over to the correct size (or at least all matching). In states which have a safety inspection, any dash lights like this would cause a failure. I'm sure if you're in Europe (UK, right?), those inspections are even more stringent (at least the ones I faced in Germany were pretty tough). There aren't many good reasons to purchase a car which has issues before you even get it home.