Shortly after getting his naval uniform, Fleming is walking down a hallway, followed by an older man in a gray suit. Fleming slows down briefly and the man in gray walks past him. In the following shot, the gray-suited man is behind Fleming, once again.
There weren't any air raids on London until after the fall of France.
In the first episode talks with his brother Peter about his fighting as a commando. In fact the commandos were not formed until much later in the war, after the fall of France.
When Commander Fleming first addresses his Naval Intelligence staff, the map of Europe on the wall behind him shows the peninsula of Istria as part of Yugoslavia. In 1939, it was still part of Italy, as it had been since 1919, and remained so until after the war.