Despite being set in 1959/1960, the Scottish Police motorcycles have 'American-style' sirens, which were not introduced until the 1980s.
Taxis shown in London and Edinburgh are the Austin FX4 model, in common use from 1958 through to the early 21st Century. However, the design of the rear lights, a long oval, was not introduced until 1968. At the time of the film's setting the rear lights were small red units, and the direction indicators were on the roof of the car, and stuck out like 'ears'.