When Prince Charles arrives at Kensington Palace in his Aston Martin and gets out of the car, the car is very slowly rolling backwards. The hand brake obviously hasn't been applied or isn't working.
After Queen Elizabeth II (Olivia Colman) asked for a meeting with Margaret Thatcher (Gillian Anderson ) following the latter's ousting as Conservative Party leader, there is a shadow of camera rigging attached to the vehicle visible on Thatcher as the car enters the courtyard of Buckingham Castle.
When the Queen awards Mrs. Thatcher the Order of Merit, she takes it out of its presentation box, in which it is stored when not being worn, and puts the box on the coffee table. Mrs. Thatcher then leaves, without taking the box.