Factual error: All of Tommy's former men salute him. As a sergeant-major and not an officer he would never be saluted. This is drummed into all soldiers and they'd never do it, even in civilian life. Not only is it against regulations, but many NCOs would take it as an insult.
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: In series 1, Sergeant Moss says he and most of his men served in the First World War, but none of them wear any medal ribbons. In series 2, Moss has suddenly acquired three medal ribbons, but they seem to be made-up and are not relevant to the First World War. From series 3, Moss is wearing the correct three service ribbons for the First World War.
Factual error: The fact Colonel Younger is black and this may be an issue socially is implied several times (meaning the series acknowledges his ethnicity), but it never seems to be seen as surprising that a black man is a colonel in the British Army in the first place. While there were a small handful of mixed-race junior officers in the army in the First World War, there were no post-war black regular officers (it was actually against regulations at the time) and no black officer would hold such a senior rank until long after the Second World War.
Other mistake: When Polly tells Ada about her two children being taken from her, she says Michael was 5 and "Sally" was 3. When she goes to the seance she says her daughter would be 18, but shortly after that Tommy finds Michael and he is 17. Also, when Tommy tells Polly that he found the records on her children, the daughter's name changes to "Anna."
Continuity mistake: Michael picks up the cash from the table. After Polly speaks to Tommy she turns around to take the money out of Michael's hand; in the shot Michael is holding the cash in his lap with his jacket off, but when Polly takes the cash it shows her hand taking the money from Michael with him wearing his suit.
Factual error: Jessie Eden is described as a shop steward. A shop steward is the senior part-time trade union official in a particular workshop or department, who also has an ordinary job in that shop. Jessie appears to be a full-time official and covers a number of factories. She would actually be referred to as a trade union organiser.