think you would have a treaty with France (Admiral Darlan was perfect barometer of German success) which strengthens their collaboration with Germany.
NO, do not mean they join Axis but the opportunity to "flip" their colonies and fleet would probably have passed and their worldwide bases are open for (at least covert) German use.
Germany had a pretty weak navy, Britain had a magnificent one. If the French colonies fought against Britain and supported the Axis they'd be picked off one by one at ease and Germany would be unable to support them. How would they propose to resupply French colonies in the Caribbean, say? Or East Africa? Also, despite what a guy in Germany said, why would somebody sat 1,000 miles away, knowing the Germans have no means of enforcing their decree, just go along with it?
So GB is going to pick off the French overseas colonies? IOTL the U.S. took the lead invading the core colonies of North Africa due to British-French tensions, here you would be creating open warfare over u-boats refueling and some smuggling of critical war materials. AND promoting the view in Metropolitan France that GB is after their colonial empire.