However, the USA entered hostilities as a response to a Japanese attack on their naval base at Pearl Harbor, 7 December 1941. Then, after the USA declared war on Japan on 8 December, Japan's Axis allies (Germany and Italy) declared war on the US on 11 December. It is unthinkable that Roosevelt would not have declared war on Japan after Pearl Harbor, whatever Edith Keeler advocated, and public opinion would not have supported anything else. Since it was the Germans who subsequently decided to declare war on the US the situation was a fait accompli: the US did not 'decide' to go to war with Germany so Keeler's life or death could have had no effect.
The only way Keeler's continued existence would have had any effect would be if she could have somehow persuaded the Japanese military to reign in their imperial ambitions before their attack on Pearl Harbor but any influence she might have had with the US President (the crucial influence according to the Star Trek narrative) would not have affected that. Without the Japanese attack the US might well have not entered the war as combatants at all.
Also, at the end of the episode, the only reason that Edith is slowly crossing the street is to see why Kirk was excited about hearing the name of Leonard McCoy. Again, if the Enterprise crew had not been in that time and place, she would probably not have even been outside at that time. Her only reason for being outside was because Kirk was taking her to a Clark Gable movie.
Only when Germany had to go on the defensive with Soviet counterattack and the United States entering the war did Hitler start giving the wonder weapons projects like the A-bomb (1942) and the V2 (1944) the funding long denied them. It such such that the best a conventional "dirty bomb" far weaker then that dropped on Japan was slated for 1946. More over the V2 simply wasn't able to carry such a weapon.
So by delaying the United States entry into WWII you ALSO delay the very factors that got the German A-bomb and V2 programs better funding.