Deliberate mistake: The Americans actually spoke with American accents, but these very new Americans were British in origin, and so they couldn't develop a new accent in such a short time. The Americans should still have British accents. This would have been confusing and is done for the viewer's benefit, but still a mistake.
Deliberate mistake: The German army is using American Sherman tanks, presumably easier to get for filming.
Deliberate mistake: Smith and the team come down the mountain and into the village at dusk, yet it is light out during the entire movie.
Deliberate mistake: The Glock pistol the dying Russian assassin attempts to fire clicks, symbolizing it's out of ammo, or had a malfunction such as failure to ignite primer which is extraordinarily unlikely. His action symbolized a dry weapon. When a modern handgun in particular a Glock runs dry, the slide stop lever engages the slide and leaves the slide open until a new magazine is inserted and the slide is racked or the slide stop is pushed down. On a dry Glock the trigger can't be pressed until the slide is cocked. (00:16:45 - 00:17:30)
Deliberate mistake: The German tanks in this movie are not really Tiger tanks. They are Russian T-34's made to look like Tigers. The T-34's are done up very realistically, including Zimmerit (the wavy pattern on the hull of the tiger), which was an anti-magnetic paste put on the tank.