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.
Character mistake: Several times Tommy addresses Sir Oswald Mosley as Mr Mosley instead of the correct Sir Oswald. Mosley never corrects him, which given his snobbishness portrayed throughout the series seems highly unlikely.
Factual error: Mosley is portrayed as forming the British Union of Fascists in 1929 or 1930. In fact, he formed the New Party in 1931. The party failed to win any seats in the 1931 general election (with Mosley himself losing his seat) and Mosley then formed the British Union of Fascists in 1932.