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.
True. I think running out of resources is a far greater threat than full scale nuclear war.
The accelerated development spike you mention was possible because we finally figured out how to use the large quantities of fossil fuels available to power factories, cars and planes.
Once it's all used up, who knows what will happen. Nuclear war to control the last coal deposits? Maybe, maybe not. Oh wait, I'm on a Star Trek forum. The future is bright, the future is bright....