The famous "Consequently this country is now at war with Germany" broadcast by Neville Chamberlain is not the archive recording of the broadcast but a modern recreation of it by an actor, made especially for the programme.
It was on September 3, 1939 that Britain, (along with France, Australia and New Zealand) declared war on Germany. The next day,the British Royal Air Force attacked the German Navy. However, on September 5, 1939, the United States proclaimed neutrality. The USA didn't enter WWII until after the attack on Pearl Harbor, December 7, 1941.
Claire Foy (Lady Persephone Towyn) & Ed Stoppard (Sir Hallam Holland) also worked together on The Crown (2016) as Queen Elizabeth II & Tony Longdon respectively.