Andrew Foyle takes off in a late war grey/green Spitfire, yet is next seen flying a dark earth/green plane. Then in the rest of the episode the late war Spit is back.
On any wartime RAF station, officers, NCOs and Other Ranks have their own messes and would never eat together, so there's no way Anthony would be having lunch with a WAAF sergeant. Modern RAF stations have an all ranks feeder which is open during the day for grabbing a quick snack meal, but they didn't happen until the 90s.
In the final airfield scene 55 gallon drums marked 'gasoline' are shown. This is an American term. This might be plausible after the US joined the war but in 1940 they would be marked 'aviation spirit' - the British term for aircraft petrol/gasoline at the time. Later Foyle's son lights a match and it illuminates a drum marked 'aviation fuel'. Again; incorrect.
Several references are made to 'RADAR', but in 1940 the British terminology was 'RDF' (Range and Direction Finding). The American term 'RADAR' may not even have been coined at that point and would hardly be in common usage since the technology was still a military secret.
After Andrew leaves the interview with the Group Captain and Wing Commander, he's seen indoors with his peaked cap on which is a breach of etiquette, caps are put on leaving the building.
Would have thought sergeants and officers dined in separate 'messes' re Officers' mess and Sergeants' mess.
Early on the man escorting young Foyle states the Germans used the radar in the Polish campaign. Many sources state Germans did not really make their radar work effectively till 1941 on wards. Hitler viewed RDF as defensive in nature and wanted offensive material. Hence later adoption. The British were way ahead with their chain home system.
The round RADAR screen with the iconic type of sweeping display was not used until after the war. The correct display was a horizontal line, as shown to the right of the round screen. Ditto the rotating RADAR antenna outside - the antennas were fixed at that time. (In any case, the antenna rotation should have the same speed as the display rotation, which it did not.)
The green Humber car HJE957 seen early in the episode was made in 1959.
Andrew Foyle is flying a Spitfire Mk. IX which was not introduced until July 1942 nearly two years after the setting of this episode just prior to The Battle of Britain. The particular Spitfire used on screen, MH434, did not make its inaugural flight until August 1943. The correct version for this time period is a Mk. I.
Towards the close, the two Foyles take shelter in a fuel store. In addition to the other errors already noted, reference is made to "highly flammable". The idiom in 1940 is "highly inflammable" (i.e. it goes up "in flames"). The more modern version did not enter common usage until the late 20th century.
The photo portrait of King George VI, famously taken by Yousuf Karsh, is seen hanging on the wall of Foyle's office in August 1940, but wasn't actually taken by Karsh until 1943.
Wing Commander Martin Keller explains RADAR and the network for dispensing that information across Britain while they are both being driven to an airfield. It's curious that WC Keller would discuss this during the drive as the driver may not have had clearance for such highly secret knowledge, especially given that Keller had just told Andrew that "...security is my [Wing Commander Martin Keller's] responsibility".
During the war women in the Air Force served in the Women's Auxiliary Air Force. WAAF officers wore a brass A (for Auxiliary) on their uniform lapels, non-commissioned ranks wore a cloth A badge immediately under the shoulder eagle. The WAAFs seen were not wearing these badges.
One of the airmen in the dining room is wearing a single chevron like a lance corporal's. This badge was not worn in the RAF until 1 April 2010, when some Senior Aircraftmen in the RAF Regiment were promoted sideways.
(In the late 1950s/early 1960s Junior Technicians wore a single chevron, but point uppermost.)