Well, before I express what could happen, I should figure out how it should happen. I think that for the Bonapartes to remain in power, Napoleon III would probably convince Russia how a united Germany could threaten the security of both nations. This would probably manage to bring Russia to France's side (technically, they were enemies since the Crimean War, but at least they have a common threat) and when the war comes, Prussia would be brought to its knees. (though I am not sure the aftermath, someone give me an idea of that)
Then, when Napoleon IV comes into power, he'll carry his father's ambition for a large colonial empire. Reduced political chaos mean more energy into building up a strong economy, military, and a strong empire.
However, it would possibly mean that France and Russia might be seen as other nations (like Britain) as a threat to all of Europe. There could be a WWI but I do not know the possible alliances or aftermath (someone give me an idea of that).