A German atomic bomb during World War II = extremely difficult but not impossible--maybe a 1-2% chance. Best bet would be having spies in Manhattan Project to help them past some of their misconceptions. That might conceivably have given them a bomb by very late in the war, but at the expense of other projects and a lot of industrial capacity. They would also have to get the project done without Allied intelligence getting wind of it. Otherwise the Allies would have hit it with all the airpower they could muster, which was very formidable any time after spring 1944.
I want to emphasize: that's for a bomb late in the war. To get one by mid-1941, the basic theoretical breakthroughs in the 1930s would have probably had to have been made more quickly. That's not impossible, but it wouldn't lead to the same Germany we saw historically having the bomb.
The first patent for an atomic weapon was filed in 1933, I believe, but there were still several practical hurdles to overcome to get to an atomic bomb. By 1939, enough theoretical breakthroughs had happened that several countries started research programs into atomic weapons, including Britain and France, which actually smuggled a quantity of heavy water out of the country under the noses of the Germans during the fall of France.
Let's try to make the best case for a German atomic bomb by mid-1941. Let's say that the Wiemar Republic military, not long before Hitler came to power, seizes on the early theoretical atomic work and decides that atomic weapons have potential to leapfrog Germany's potential enemies, getting around restrictions on rearmament by creating a superbomb.
The German military has been carrying on secret research on a variety of weapons, including tanks and aircraft anyway, so this is just one more clandestine project. They secretly enlist Germany's first-class scientists, or at least some of them, and give them as much support as they can given Germany's slender financial resources. The program is very secret, of course, and not huge initially. It's mainly theoretical and experimental work at this stage--not cheap but not huge, not on an industrial scale.
For this to work, the Germans have to make very quick progress, push probability to the extreme edge of what's barely possible type progress. Let's give them that, for the sake of argument. By the time Hitler consolidates power in the night of the long knives, two things have happened: (1) The Germans have made enough progress to know that there is an enormous amount of potential for a weapon here, and (2) In the small community of theoretical physics, the absence and secretive behavior of German heavy-hitters has become known to the rest of the physics community. Non-German physicists become aware that something secretive and military is happening in Germany.
The German military at first kept quite a bit of autonomy from Hitler, but as the program progresses, Hitler becomes aware of it. His overwhelmingly most probable reaction: to try to disband it because of the many Jewish physicists involved. Let's say that somehow the army convinces him of the potential for the bomb, or manages to ignore his directives initially. The first of those is extremely unlikely, the second quite possible in the early years of Hitler's reign. So the program continues through the mid-1930s. By this time (1) the military potential has become quite obvious, but (2) Persecution of Jews has gotten bad enough that Jewish (and some non-Jewish) scientists on the project are becoming very reluctant to work on it. The Germans start facing subtle sabotage, and quite likely defections or attempts to leak what they're up to.
By 1937 or so, the rest of the world would be getting quite alarmed at the possibility of a German atomic program, based on the silence of the German physics community, and possibly leaks from it, or key defections. The possibility of an aggressive Nazi Germany with superbombs is a nightmare for the rest of Europe. Every other major power quietly starts their own program, racing to catch up with the Germans, with some of them helped by leaks and/or defectors.
Meanwhile, in Germany, leaks or defections from the nuke program could precipitate a showdown between Hitler and his generals. Even without that, Germany would have to reach a decision point by 1937 at the latest: do they pump scarce money and industrial resource into building the industrial infrastructure for bombs or do they use those resources to continue their conventional arms buildup. That's a tough one. There is a good chance that they go for conventional arms. Let's say they go for a nuke industry instead, taking the resources out of their conventional buildup--so fewer planes, fewer ships, and fewer tanks.
By the time Munich rolled around historically, the German conventional army is in no shape to take on even the Czechs alone, much less the French and Brits. But Munich was largely a bluff on Hitler's part anyway, so maybe even the reduced German military is enough to bluff the Allies into abandoning the Czechs. That's not all that likely. By now the evidence of a German nuke program would be getting pretty overwhelming and the Czechs have uranium that the Brits and French would not want to have fall into German hands.
Assuming that the W. Allies still don't fight and the Czechs don't decide to fight on their own and kick weakened German butt, that takes us to 1939, where the Germans are still two years from a bomb, but much weaker conventionally. Do they risk war in Sept 1939? Probably not. Their conventional edge over the Poles would be enough smaller that they couldn't expect a quick victory. Also, the threat of a German atom bomb would probably push the W. Allies and the Soviets together. It would also probably mean a less isolationist US.
Even if the Germans managed to take Poland quickly, they would not be in a position to take France. Even to do the conventional arms buildup they did historically, the Germans had to keep taking and looting new countries. If they hadn't taken Austria they were facing 30% cutbacks in military spending, because they were running out of money. Looting Austria tided them over until they got Sudetenland, which tided them over until they grabbed and looted the rest of Czechoslovakia. They had to keep getting easy victories, and that's even without a nuke program devouring resources.
See how all this works? A German a-bomb by 1941 wasn't totally impossible, but it runs a gauntlet of improbabilities and ends up with the Germans (a) in a much weaker position economically and in terms of convention weapons, and (b) Probably not very far ahead of Britain, France and others in terms of their atomic program.
. A wikipedia stub with broken links isn't an serious article.