Factual error: Bill wears four medal ribbons, in order: Diamond Jubilee Medal (awarded to all people with five years' service in a British uniformed service in 2012), Golden Jubilee Medal (the same in 2002), Queen's Police Medal (a very prestigious award, a handful of which are awarded every year to long-serving police officers for exceptional service) and Police Long Service and Good Conduct Medal (awarded for 20 years' unblemished service in the British police). Not only are these in the wrong order (it should be QPM, Golden Jubilee, Diamond Jubilee, LS&GCM), but he's not entitled to any of them! He's an American who hasn't previously served in the British police. A chief constable's uniform doesn't just come with medal ribbons for the look of it; only those that have been earned may be worn.
Audio problem: When Michael is forced to do "errands" for Brennen, they stop by Jonathan Carver's house to get numbers from him for a voice key. The code played back at the facility is 71348. He says "3" in a medium pitch (tone) at the house, but it is high pitched when played back at the facility; he says "8" in a medium pitch, and it's also high pitched later. And finally he says "9" in a low pitch and it's played back medium pitch. Also very obvious is the fact that Jonathan never says "1" or "7."
Other mistake: Crowley shows acute discomfort while inside the church with Aziraphale and the Nazi spies as he is on consecrated ground. However, (spoiler alert) once the bomb destroys the church he is perfectly comfortable. The key word here is "ground" - it is the land upon which the church stands that is consecrated, not the building itself. (This allows churches to be rebuilt if they are demolished or destroyed.) Crowley should still be in considerable discomfort after the church has been blown up.
Revealing mistake: Watch the opening sequence of this episode very carefully - Clarkson, May and Hammond are not in Downing Street (they are on a set that stands on the BBC backlot) and library footage of the Prime Minister is cleverly edited in to make it look as if they were there at the same time. Even taking into account the fact that the Prime Minister's security service would never allow such an intrusion onto Downing Street, the line up of the shots does not quite match - our three intrepid heroes should be visible in shot when the Prime Minister is shown exiting number 10, but they are nowhere to be seen. It is all sleight of hand and clever editing, but detectable if you watch carefully enough.