Factual error: In the House of Commons, Tommy refers to an unseen Tory MP as "My Right Honourable Friend" and as "the member for Epping." The former address is only used for Privy Councillors of the speaker's own party (Tommy is a Labour MP); MPs of other parties would be "the Right Honourable Gentleman." Winston Churchill was MP for Epping at the time and Tommy's secretary later says Churchill is in America, so he can't possibly be addressing the member for Epping.
Factual error: Several characters refer to "the London Times." It's never called this in Britain. It's just "the Times."
Factual error: Polly refers to Ada's soon to be born child as "the colour of a Hollywood Oscar." The first time that the statue was referred to as an Oscar was in 1939, so Polly saying that was 10 years too early.
