Does The City on the Edge of Forever claim that the Germans could have built a bomb in our timeline? No, the claim is that with America out of the war until the Germans completed their heavy water experiments, Germany could have had such a bomb.
The dialog between Spock and Kirk doesn't really make much sense:
SPOCK: This is is how history went after McCoy changed it. Here, in the late 1930s. A growing pacifist movement whose influence delayed the United States' entry into the Second World War. While peace negotiations dragged on, Germany had time to complete its heavy-water experiments.
KIRK: Germany. Fascism. Hitler. They won the Second World War.
SPOCK: Because all this lets them develop the A-bomb first. There's no mistake, Captain. Let me run it again. Edith Keeler. Founder of the peace movement.
KIRK: But she was right. Peace was the way.
SPOCK: She was right, but at the wrong time. With the A-bomb, and with their V2 rockets to carry them, Germany captured the world.
The problem is how could Edith Keeler have prevented Pearl Harbor the real trigger for the US entry into WWII?
Ever in OTL the Germans were meh on developing an atomic bomb so how did Keeler's peace movement in the US make the German more aggressive?
Sure it is good drama but in terms of actual history it sucks.
President Roosevelt moved the Pacific fleet from San Diego to Pearl Harbor in early 1941, as a rattling of our saber to the Japanese to signal that they should watch their step in the Pacific. It's not hard to imagine that, under the influence of Edith's peace movement, the President would have chosen to keep the fleet at San Diego. That would have eliminated the opportunity for the attack on Pearl Harbor. A stronger and more overt pacifist tendency in the United States also would have reduced the motive for the attack, which was to keep America off Japan's back while they expanded their empire in the Pacific.
As Frank Capra's Prelude to War shows there already was a strong isolationist-peace movement in OTL.
Furthermore in OTLthe USS Panay and Nanking Massacre of 1937 resulted in the US (along with the UK and France) providing loans to China. In 1940 things got to the point that many of the exports form the US Japan needed to keep the whole war machine were cut off and in 1941 oil followed.
The Japanese military believed that the ties between the US and UK were such that any attack on the British Southeast Asian colonies would bring the US into the war and they needed to take those colonies. In fact the attack was supposed to occur 30 minutes after war was declared but miscalculations in the sending of the message and its decoding resulted in the Day of Infamy.
You also must remember that in the 1930s fear of the yellow peril (Fu Manchu, Ming, etc) was very real and it was easy to transfer that fear from the Chinese to the Japanese. In fact in 1940 the U.S. State Department actually asked that no more Fu Manchu films be made and once the US went to war Rohmer found himself unable to publish any additional Fu Manchu stories.
As to the effect of America's absence from the European theater until much later, you are looking at it all wrong. Of course that didn't make the Nazis more aggressive; they were already brutally aggressive. But, as I've said twice now, what it would have done was make available more resources for them to spend on wonder weapons. There is general agreement that the Nazis spent more on their rocket program than the Americans spent on the Manhattan Project. I believe a level of additional expenditure less than their rocket program to put towards a wonder weapon would have been within the means of the Nazis, if the Allies were not bombing at the levels that they were, taking territory on the Western Front, and pressing towards Germany, which all necessitated immediate and costly defense by the Germans. This change in strategy no doubt could have increased German's security on their Eastern Front as well.
This ignores Mark Walker's point "From the start of the war until the late fall of 1941, the German "lightning war" had marched from one victory to another, subjugating most of Europe. During this period, the Germans needed no wonder weapons."
Look at the Vergeltungswaffen weapon time table:
Fieseler Fi 103 (V1) - June 13, 1944
Aggregat-4 (V2) - 1944
Both partly developed in response to the US getting in the war. Remove the US as a player and no pressure to push for these or any other wonder weapons.

It's a Catch-22. For Germany to be motivated to produce the wonder weapons needed the US must become involved in the war and development on those weapons only starts in earnest after the US gets involved--by which time it is too late.
A more reasonable AH would be that the delay resulted in the Soviet Union and Nazi Germany effectively bleeding each other nearly dry so that the Allies decided that taking out both Fascism and Communism was a great idea. This would result in no cold war and no push for a moon landing leaving Earth far less technologically developed then needed when the Vulcans fly by.