Something Star Trek tends to do is measure technological process by years. This carries the implication that the amount of time something takes on Earth is the exact amount of time the same amount of technological development will always take.
Look at all the hyper-accelerated development we've had in the past few decades, after millennia of far slower paced technological development. That kind of spike only needs two ingredients: The ability to finely control electricity and the economic infrastructure to develop it. That doesn't have to happen at a specific number of centuries after other pre-electric milestones.
I don't think it's impossible that we will have a massive nuclear war, but I do think that the odds are lower than you say. Most evil people would rather control everybody than kill everybody.