I know this is a necro, but I only just found this thread.
A lot depends on just how powerful Germany is, between 1940-60. Are we talking about a Germany that has crushed Russia in 1941 and now lays claim to everywhere from Dunkirk to Kamchatka, or a Germany that made peace with Russia in 1944 and has far less of a free hand in post-war Europe? If the former, the French have relatively little bargaining power; if the latter, they have a great deal more because the Russians will be permanently on the borders, ready to attack of the regime looks weak.
Hitler, IIRC, hatred the French and saw them as permanently inferior to the Germans. This might change if it became obvious he needed French help, or (at the very least) he needed the French to stay quiet rather than turning into a major distraction at the worst possible time. Ideally, from his point of view, Vichy would be a puppet state – allowed a little independence in exchange for servitude. This would basically force the French to bow down indefinitely or secretly rebuild, planning revenge. Hitler would be very aware the French got hammered in 1871 and still managed to come back for a rematch in 1914-18, which the Germans lost.
I suspect the Germans will keep a close eye on the French, to make sure they're not rebuilding their army to dangerous levels. The French will probably resist allowing Germans into their colonial empires, both because it will be harder to rearm if the Germans are poking around and because it would make them look weak in front of their subjects. (Although the idea the subjects don't already know strikes me as a little silly.) There might be a split between the Free French, if they exist in this world, and Vichy; the best and brightest of France, aware they won't be allowed to rise in the Nazi-dominated world, will sneak away to the Free French. There might be a very weird arrangement between the two French factions – the Free French drain away the most dangerous youths in Vichy, rather than letting them grow up discontented.
Long-term, Vichy might well lose control of much of its empire. Revolts were a fact of life across the empires and the French wouldn't be able to muster a really powerful army in this timeline because the Germans would object strongly, assuming the French secretly planned to give their men experience and then make a bid for independence. On the other hand, if there is a second exodus of Frenchmen to Algeria (there was an exodus after 1871, IIRC), there would be a much larger French population in Algeria and perhaps a different outcome to the war. Vichy might also learn the wrong lessons from the Germans, who'll have their own problems in Eastern Europe; they might mount genocidal campaigns rather than try to separate the educated (and presumably pro-France) Arabs from the rest.
Really long term, if Germany collapses al la the USSR, the French may regain their independence. But would they be able to repair the damage of years as a puppet state?
Chris