When Charles goes into Sarah Spencer's house he's wearing shoes. Next part of scene riding boots and then back to shoes again.
In this episode and for the remainder of season 4, The Queen's private secretary is still Martin Charteris. In reality, Charteris retired as private secretary in 1977 and entered the House of Lords as a life peer and was succeeded by Philip Moore. Moore remained in the position until 1986 before being replaced by William Heseltine.
When Philip is shooting pheasants his gun does not recoil.
When Charles asks Sarah for Diana's number she states the area code as 01373, however the UK only changed area codes to include an extra "1" after the first zero on PhONEday (16 April 1995, the capitalization of ONE in the name was intentional), before that the code for Frome was 0373.
However, 01 was the original dialing code for the whole of London and 373 was the exchange code for Earl's Court - the area where Diana lived at the time.
Near the end when Charles telephones Sarah to ask more about Diana, the telephone ring is a US one - like the ones commonly available on current mobile phones - it's a equally spaced single ring and gap. UK phones with a bell at the time rang twice before each gap.
The assassination of Lord Mountbatten took place in 1979. The montage of images for his funeral show scenes that happened later in history. For example, Bobby Sands and the names of other hunger-strikers are shown, but the hunger strike was not until 1981.