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.
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: 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.
Factual error: When Freddy and Ada return to Birmingham from London they are seen on a boat called Kennet. This is a wide boat (i.e. almost twice the width of a 7 ft narrowboat) which could not have made its way from London to Birmingham in 1919/20 as all of the locks were narrow locks (7 ft) at that time.