When the rocket is shown in the rec hall and shown being wheeled in the compound, it is only about 12 feet long. When it takes off, it is a much bigger rocket.
The rocket that is seen taking off from the rec hall does not have fins. Seconds later, the rocket shown in flight has large fins.
The wheels on the cart were small when it was taken out the dog house. They are larger when the cart is being wheeled around.
They mention pilots for the V-3 rocket, when it was actually a cannon, and neither the V-2 nor V-3 had any pilots.
When the rocket is launched from the Recreation Hall, the exhaust flames from the rocket would have completely destroyed the entire building - even with the little deflector plate that was in place on the makeshift launch pad.
When Hogan and his men are first examining the rocket in the woods, you can hear crickets chirping in the background. Since it is Winter and there is snow on the ground there is no way the crickets would survive in those conditions.
There are heavy shadows on the wall in the scenes in the rec hall. These are shadows from the production lights.
A rocket of the size shown would not be large enough to fly from where Stalag 13 is located in Germany to either England or Moscow.
There is a production light reflection in Klink's monocle.