It certainly didn't help that the Nazis had a brain-drain in physics because of their anti-Semitic policies, no. But that wasn't why they chose rockets over atomic weapons.

You have to remember that at the time they made their "choices" for these programs — e.g. around 1941-1942 — they were doing pretty well in the war. They didn't see it (as the Allies did) as a "race to get X before we are defeated" sort of thing, which is what motivated the Allied atomic work. For the Germans, nuclear fission looked like something to think about for the next decade not the next two years, which is more or less when they thought the European hostilities would be done by (in their favor). So they approved a modest reactor program but not a "crash" effort.

The rockets on the other hand looked achievable in the shorter term (the V-1 and V-2 started being produced in 1942, used in 1944-1945). And while they certainly proved to have no effect on the war effort, it wasn't clear to anyone initially whether they would be powerful weapons of propaganda. (It wasn't clear about the atomic bomb, either, for that matter!) What do you do if you make a super weapon and your enemy just shrugs it off? World War II was a proving ground for a lot of different theoretical ideas about how strategic attacks worked (e.g., could you "break the will" of a nation by bombing them with increasingly sophisticated means) and both the Allies and Axis had ideas that experience proved to be false.

Anyway, here's the crux of your question: could they have developed an atomic bomb if they had made the decision to do so? The odds are no, at least not before the war ended in Europe. Why not? Because building an atomic bomb requires large, detectable facilities. And the Allies were specifically looking for facilities like that and bombing them. This is a fundamental asymmetry between the Allies and the Axis by the mid-to-late war in Europe: the US enjoyed nearly total impunity to build and manufacture whatever it wanted, on any scale, because the oceans essentially made it invulnerable to attacks against its production facilities. The Germans, on the other hand, suffered increasingly heavy bombing attacks as the war went on. And this makes nuclear programs in particular an unlikely endeavor, because the facilities are highly specialized and hard to rebuild.

Keep in mind that even the United States struggled to produce a nuclear weapon by the end of the war with the Japanese, with as much funding and labor resources as they could ever want, in a situation where the bomb project was given literally the highest priority rating of any military production, in a world in which the massive facilities, employing hundreds of thousands of people, were spared from hostile attack or sabotage. Even under conditions that are remarkable even from the present day (and indeed the Manhattan Project is still the record-holder for "fastest time to a bomb once the decision to build one was made," 1942-1945 = 3 years), the US couldn't make a bomb in time for it to be useful in Europe, and if things had dragged out just a few more months, may not have had one before some kind of surrender was worked out with the Japanese.

The idea that the Germans, under their conditions of relatively smaller resources (both financial, materiel, and human) and, as the war went on, increasingly insecure production capability (thanks to bombing) could have somehow pulled off a faster bomb than the USA did is hard to fathom.

Going to zero in here on the "useless" V-2 program comment, since /u/restricteddata gave a good overview of the overall question.

Could the Germans have produced more planes had they halted production on the V-2? Probably. But by September 1944, when the V-2 began combat operations, the Germans faced a critical shortage of trained pilots. They were overwhelmed in the air; the Luftwaffe had its best pilots fighting a last-ditch defensive war while Germany's cities were being pounded into splinters. They didn't have the luxury of dedicating their best pilots to non-combat postings, where new trainees would spend months in the air before their first taste of combat. The German situation was desperate. By any reasonable standard, the war was already lost, and reasonable methods would not turn the tide.

The V-2 allowed the Germans to shift the strategic balance, if only slightly. It restored the ability for Germany to strike behind enemy lines, at centers of strategic importance like Antwerp and London. It gave Germany a shortcut past Allied air superiority. It forced the Allies to slow their advance and allocate resources to defensive measures. Most critically, the V-2 didn't require Germany to squander more trained pilots, and it used an alcohol-based fuel, not desperately needed gasoline.

Hitler himself was skeptical of the program at first, but allowed himself to be convinced. Once it began successful attacks, the Nazi government seized on the V-2 as a "wonder weapon," of course. The rocket program is not a distraction, or an oddity, but a natural outcome of Hitler's proclivity for seizing on gambles that seem reckless in hindsight.

After all, the war itself was the largest of Hitler's gambles. His apocalyptic worldview spanned two futures. In one, Germany pushed aside its neighbors to establish a continental-scale civilization with the population and resources to compete with the world's largest and most successful- China, India, the United States. This is the true meaning of the "Thousand-Year Reich." In the other, Germany would succumb to its natural racial and ideological rivals. Given the all-or-nothing struggle Hitler believed was already underway, what was the gamble of loading up Germany with debt equaling 120% of its GDP (in just six years)? What was the gamble of invading the Soviet Union? What was the gamble of throwing everything on the Western Front into the Battle of the Bulge, declaring war on America, invading France, allowing Dunkirk, starting the V-2 program?

Hitler believed the war was underway and likely lost, even before he started it, and that only his singular vision could win it. His seemingly bizarre decisions make a lot more sense in this frame.

Hey there,

Our experience as moderators on /r/AskHistorians is that questions framed as 'Why didn't X do Y' often don't get a good response. There are a few reasons for this: firstly, it often can be difficult to prove the counterfactual - historians know more about what happened than what might have happened. Secondly, these questions are often fundamentally ahistorical - people in the past could not see into the future to know which plans would work, and they very often did not know all the information about the situation they were faced with that we now know with the benefit of careful work by historians. Thirdly, some historians feel that these questions unfairly assume that people in the past were a bit dumb, and this might make them less likely to answer your question. We're not removing this post - you might still get an answer! - but if you want us to remove this question and ask a similar question with a different angle, let us know and we'll remove it so you can ask the new question. Hope that helps!