Although set during WW2, almost all the characters (including the women) wear 1960s hairstyles. Regulation army haircuts would never have permitted sideburns or hair long enough to touch the collar or go past the ears.
When the soldiers push the VW into the moat, the passenger side window is open. The car sinks and then pops to the surface and floats (air tight and they really do float) ... but when it surfaces, the passenger window is up. Of course, the car would have stayed on the bottom, had the window been open.
The major switches uniform types during the movie. From regular infantry and then later to Airborne. This wouldn't happen. Airborne uniforms were limited to airborne troops and would not be available to others.
In a very typical mistake for the period, the "German" tanks are all ex-Soviet T-34-85. German tanks were simply unavailable, but no attempts were made other than painting them Grey, which was also incorrect for the period.
The soldiers refer to the car as a "Volkswagen", a name which some viewers believe was not given to the car until after World War Two. However, on 28 May 1937, the Gesellschaft zur Vorbereitung des Deutschen Volkswagens mbH was established by the Deutsche Arbeitsfront. It was later renamed "Volkswagenwerk GmbH" on 16 September 1938. It is not clear whether the GI's would have easily recognized the car, but any German would have readily recognized it as a Volkswagen during the war.