Baron von Rheingarten is wearing a forage cap in the beach scene. In the next scene when arriving at Door's house he is wearing a peaked cap.
When Bernhardt is walking in the square and asking to have his film developed, he says, "Can't go into the shop, if I do I'll be sent to the eastern front." There was no eastern front in 1940.
Philip Dorr says the capture of the Channel Islands was the first invasion of British soil for 900 years. The French briefly invaded mainland Britain at the Battle of Fishguard in 1797.
After the photography shop has been branded 'Jewish' by the occupiers, Tellemann tells June that he can't be seen going there, or he'll sent to the Eastern Front as punishment. But Episode #2 is set during the summer of 1940, when the Germans were not yet at war with the Soviet Union.
Pennsylvania 6-5000 was released in the Unites States in June 1944 and the first records did not reach the United Kingdom until August. The Germans would not have known the song in 1940.
Von Rheingarten is playing jazz when the Dorrs join him for brandy. The Nazis considered jazz to be degenerate and it was banned.