Initially all cars, road or track, were right hand drive. The pre-1905 cars that take part in the annual VCC London-Brighton run are either centre or right-hand drive irrespective of country of origin, well over half are from outside the UK.
According to Wikipedia, it was Henry Ford who first put the steering wheel on the left - in the Model T - after which other manufacturers gradually followed. As Peter says, Lancia was one of the last, persisting with RHD into the 1950s.
Driver Seating PositionOne of my favourite facts about LHD / RHD is that the U.S. Virgin Islands (a former Danish colony) is the only American jurisdiction that drives on the left. However, almost all the cars there are sourced from mainland America. Consequently, American tourists in LHD hire cars give the territory one of the world's highest road accident rates.