Deliberate mistake: David is shown to have an illegal weapon in his home even though gun control is quite strict in the U.K. Sadly, this is quite a common practice by returning war veterans who feel safer with a gun in the house, as they often suffer from PTSD, as David does.
Character mistake: Budd is constantly shown buttoning up his jacket when he exits the limousine prior to Julia exiting. As a PPO, SOP is to unbutton to make reaching his firearm easier, not harder! Also, when he buttons, you cannot see the firearm on his belt at all, even though the jacket is quite tight.
Episode #1.6 - S1-E6
Trivia: In the park where PS Budd is surrounded by armed tactical response officers, a careful observer would notice that the brand of ear protection devices the officers are wearing are Z-TAC, an airsoft brand.