Napoléon III would likely expire during such a war, he wasn't in the best of health IOTL as is. As well remember that our POD is already after Sedan, so he's already surrendered his person to the Germans (as opposed to his empire; though that's a legal distinction few will pick up on). This technically leaves the empire to Prince Napoléon, whose just a young teen as yet, under a regency by his mother Eugénie. Unfortunately for the imperial cause both of them had fled to England following Sedan. As well, it was the Government of National Defense ITTL that repulsed the Germans (assuming a French victory at Paris). Now, the situation is certainly chaotic enough that things could change if the Prince Imperial returned to France immediately, preferably without his mother, as a) she had plenty of her own enemies but he could be a tabla rasa, and b) he needs to show he's 'his own man.'
As for the Germans, I doubt that the lower German states unify with Prussia's North German Confederation into any sort of imperial arrangement. For one, this war is a draw at best, as opposed to IOTL's quick victory. That's a massive change. As well consider that ITTL it was the Bavarian, Württemberger, Badenish, and Hessian commanders that called for drawing the French out of Paris, or starving the city, as opposed to the Prussian demand for an immediate assault. They will (rightfully) conclude that Bismarck and Wilhelm goaded the French into attacking, and then even more foolishly attempted to storm their own capital with a larger force behind the city walls. These type of things are easily forgiven, or forgotten. Berlin simply doesn't have the prestige among the German states that it did IOTL to pull off bullying the other few remaining German states into itself.