As the horse runs down Alec's street with Alec chasing him, after the red car pulls over, a contemporary (i.e. black with white background) "Speed Limit 25" sign can be seen. Most speed limit signs then were black background with white lettering and would have read something like "Speed Limit 25 Miles".
When the horse is frightened by trash collectors, it is seen to run down a lane. There is a red sports car parked at the end. The car is an MG model TD, which was put into production either late in 1951, or early 1952. It looks period, but isn't.
When Alec first befriends the Black on the beach using dried seaweed, he runs away from the Black so the Black will chase him, and in alternating scenes during the chase he is sometimes carrying a handful of seaweed and sometimes not carrying any.
Just before the start of the final race, the camera gives a view from the cage to the main sector for visitors (at the right upper corner), which seems absolutely empty, even though the stadium is overcrowded in the previous and the following scenes.