The Japanese are rather fond of America, if I am not mistaken. I assume that it is not uncommon for the conquered to eventually grow satisfied with their conquerors.

For the Poles at least I fear it probable not many would be around say 20 years after victory.

It's irrelevant. In a world of world-destroying technologies, a really bad thing happening for only a small amount of time is all it takes. The Cold War wasn't even close to the horror of Nazi domination (probably)--there were still lots of happy people with decent governments in the west! But everyone still could have died.

What if Nazis had developed nuclear weapons? What if the AI self-reproduces, without self-improving, such that the Big Bad they're supporting has an army of super-efficient researchers and engineers? What if they had gotten to the hydrogen bomb around the same time the US had gotten the atom bomb? What if the Big Bad develops nanomachines, programmable to self-replicate and destroy anyone who opposes, or who passes a certain boundary? What kind of rebellion or assassination attempt could stand up to that? What if the humans want the AI, rather than another human, to be the leader of their Big Bad Movement, making their evil leader both easily replicable and immune to nanomachine destruction?

Hell, what if the AI gets no more competent or powerful than a human? It can still, in the right position, start a thermonuclear war, just the same as high-level weapons techs or--hell!--technical errors can. Talented spies can make it to sufficiently high levels of government operation; why couldn't a machine do so? Or hire a spy to do so?

And if the machine thinks that's the best way to make people happy (for whatever horrible reason--perhaps it is convinced by the Repugnant Conclusion and wants to maximize utility by wiping out all the immiserated Russians), we're still in trouble.

However, if you're trying to describe an AI that is set to maximize human value, understands the complexities of the human mind, and won't make such mistakes, then you are describing friendly AI.

Edit: In other words, I contend that the future threat of General AI is not in modifying humans with nanotechnology. It is in simple general ability to shape the world, even if that only means manipulating objects using current technologies. If we're defining "intelligence" as the ability to manipulate atoms to shape the world according to our bliss points, a machine that can think thousands of times faster than humans will be able to do so at least hundreds of times better than humans. This is especially true if it can replicate, which, given this hypothesis, it will almost certainly be able to. If we add intelligence explosion to the mix, we're in big trouble.