When Budd looks up the Home Secretary's voting history, there is a sentence that shows she consistently votes in favour of UK military action overseas, which ends with a period. When the camera zooms in on the sentence, there is no ending punctuation.
Very early in the first episode we see David put on a white shirt. The shirt features later in the episode, but when he is next seen, it is blue, then white, then blue and then white again.
The train is being pulled by a British Rail Class 47 in the British Rail blue livery. The Class 47 had gone out of regular passenger service in the 1990's. Also British Rail ended in 1997 so the livery is also obsolete. Lastly, the MK3 Coaches hauled by the 47 are now only ever hauled by excursion trains, The Class 43 HST and the Class 91.
Towards the end of the episode, David loads a pistol. He retracts the slide -- and then guides it back forwards. Guiding the slide/bolt of a self-loading weapon back forward is not only unnecessary, it can cause a malfunctioning. Anyone who went through basic military / law enforcement training should know this -- and certainly highly trained bodyguards.
After David's kids exit the police car, his son runs a couple steps too far past the sidewalk entrance to his house and quickly realizes his error and runs back towards his mother and sister.