Character mistake: Referring to the US, Russian and Chinese governments, the admiral says that "Washington, Moscow and Shanghai" have been informed. The capital of China is Beijing.
Factual error: Many of the Royal Navy details are completely wrong. The submarine captain's cap badge is not that of the Royal Navy. "Midshipman" Leonard wears the insignia of a chief petty officer. "Officer" (not a naval rank) Neal wears the insignia of a petty officer. "Officer" Shaw wears the insignia of an able rating. The Flag Office Submarines is depicted as a full admiral, whereas in fact he was a rear-admiral and the position no longer exists.
Character mistake: The submarine captain says they have no "kitchen staff" on board. No naval officer would refer to a ship's galley as a "kitchen". It's a tradition thing.
Factual error: Matt says he was in the Royal Marines, to which the submarine captain says "you were in the army?" Any naval officer would know the Marines are certainly not Army.