If you accept the premise that Germany gets the bomb first, and is therefore able to keep the rest of the world under control through fear of it, I don't think it is so ridiculous. It is not as if the conquered populations accept things totally without question, there is resistance after all.Given it is a blockbuster TV show it is not nearly as ASB as one might expect.
The issues I have with it are comparitively minor ones, one being how Christianity has been banned (doesn't fit with what we know about Nazi relations with the Church, and unlikely to be accomplished without major unrest) and the other being the confusion surrounding precisely how the satellite states work. It is implied that the eastern half of America is a Nazi client state, but it seems to be to all intents and purposes run directly from Germany, there is no mention of any American leadership, and the people charged with keeping the peace dress exactly like Nazis, they even use German titles.