The girl's parents and brother were taken to hospital and their injuries were not thought to be serious.
The male driver of the Peugeot was arrested and taken into custody. A female passenger was taken to hospital suffering from minor injuries.
The incident happened just before 6.30pm and police, ambulance and fire crews attended.
The Metropolitan Police said the pursuing police car had its blue lights and sirens on when the collision happened.
More top news
The journalist reached an agreement with the corporation last month.
The Russian leader spoke during a press conference following a summit with US president Donald Trump in Helsinki.
Leading former anti-apartheid campaigner Lord Peter Hain, who is chair of the Nelson Mandela Centenary Exhibition, welcomed the royal suppor