Fascist regimes will never accept even a mutually assured destruction status quo. Any fascist regime with nuclear weapons will be a regime that uses their nuclear weapons. There is no cold war in such a situation, only the brief pause before the fascists get the arsenals they feel are necessary.
The Germans were well ahead in the development of chemical weapons but they thought they were actually at parity with the Allies and refrained from using them. Germany also refrained from using chemical weapons because it would have risked contaminating friendly forces and disrupting logistics for their maneuver warfare. On a more strategic scale, it also doesn't make sense to devastate land desired for resettlement by using long lasting weapons such as chemical and nuclear weapons.
Another issue is that you are assuming that the scientists working on the program actually want it to succeed. If they think that the weapons are actually going to be used, especially for genocidal purposes, they might deliberately slow things down. This is something Werner Heisenberg may have done with the German nuclear program.
The Commonwealth/United States and Soviet nuclear programs didn't have those problems because they were working under the assumption that Germany was well ahead and wouldn't hesitate to use them first. As the Cold War developed many of those same scientists tried to slow the arms race. J. Robert Oppenheimer opposed the development of thermonuclear weapons and Andrei Sakharov later became a prominent dissident due to the implications of his work.
Because fascism values things like the cult of action and race over their own lives.
Apartheid South Africa is the only state in history to abandon a domestically developed nuclear weapons capability. To clarify, I don't mean that they were working on nuclear weapons and gave up, I mean they had a factory producing Little Boy gun type nuclear bombs and had aircraft capable of delivering them.
The South African nuclear weapons program wasn't just the creation of the Apartheid state, but intended as an insurance option for it. Instead of using the nuclear option the Apartheid government decided to dismantle the program (without so much as a confirmed test) and begin the transition to democracy.
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