When Carmine confronts Alicia about the communicator he found in her jacket, he says it looked like the thing Archer took off 'those Martians'. In fact, the two props look very different and a mind raised in the early 20th century would not have made that connection.
When Archer and company are fighting the Germans at Vosk's facility (in New York) Enterprise is seen approaching the East Coast from orbit. As seen from Orbit New York is already seeing the sunrise, but on the ground it is still completely dark.
The Enterprise is attacked by Stukas. Stukas, short for "Sturzkampfbomber" = "Assault dive-bombers" were never designed for aerial combat; their role was to eliminate ground targets - the planes were way to cumbersome and could not have performed some of the maneuvers shown. Particle weapons notwithstanding, the likely plane for this role would have been the Bf 109 or the Me 262.
Reed reports that in the altered history, the Nazis did not invade Russia, but only expanded westward. Yet in the previous installment Star Trek: Enterprise: Storm Front, the German General told Vosk that Germany's forces are strained because the Russians are attempting to retake Moscow.
Archer suggests Silik needs Enterprise to send the information gained about time travel technology into the future, but that is not true. Time travels forward all on its own. All Silik would have needed to do is bury a few discs somewhere safe on Earth. Then, his employer could have another recruit dig one of them up at an opportune moment later in time.