At the end of the film, the Royal Family goes to Saint Paul's Cathedral. A view of the front of the Cathedral shows that the clock in the left-hand tower is missing, but this was as a result of German bombing raids in the early 1940s.
During all of the live musical performances (most notably during the George Frideric Handel Water Music performance that the King attends) the instruments are tuned to the modern tuning of A=440Hz instead of the lower pitched A=415Hz that 18th century instruments were tuned to.
The servant Fortnum states that he is leaving royal service to open a provisioner's shop in Piccadilly ("a step up from emptying piss-pots"), a comical allusion to the Fortnum & Mason's establishment. The film takes place in 1788, 81 years after F&M was founded in 1707. In fact, the company's co-founder's grandson entered royal service in 1761, which led to an expansion of the company's business.
Pitt and Thurlow go to visit the King at Kew during the winter - but the leafy green trees outside clearly indicate that the scene was filmed in summer (shooting took place from July to early September).