Each colony had its own reason for independence, but to generalise...

Vichy France assumed control over the majority of colonies it still held so life of an average citizen in say modern day CAR did not change a great deal. Plus there were still a great deal of Axis forces within North Africa at the time.

The Allies not long after the establishment of Vichy France then launched an invasion of North Africa, Operation Torch, so between the 2 armies battling it out in the North there was little a group of rebels could have done. They could not exactly take on both the Allied and Axis, both of whom wanted control.

Finally, there wasn't a great call for independence during the war. That came after. Many future Africa leaders were in Europe during the War and the Cold War helped, with the superpowers reluctant to see the other claim a new ally. Plus there was a growing reluctance in France of maintaining an Empire when France itself was ruined.

I hope this sort of helped. Its a little general, but hey.

It's worth noting that France's colonies in Southeast Asia were occupied by the Japanese during the war. Indochinese nationalists fought the Japanese and then and then continued to successfully fight the French after World War II ended. Look up on the Viet Minh and French Indochina during the war for more info.

There is a kind of important detail that has been left out so far.

Vichy France was the legal government of France throughout the war.

It was deeply collaborationist but it was not installed by the Germans. The Vichy French government was formed when the Third Republic appointed Marshall Philippe Pétain as head of the new government to make the most favorable peace with the Germans he could. Anyone breaking away from Vichy to join the Free French was breaking away from their legal government.

The French government wasn't in exile, the French government was the Vichy France regime. A 'Free French' government-in-exile was of course set up by Charles de Gaulle in London, but was always kept very firmly under the thumb of the British who had a very frosty relationship with de Gaulle.

Churchill's government did indeed did put pressure on the French to declare their Middle Eastern mandates independent (presumably to open them up to British influence), but de Gaulle rather smoothly turned the tables on the British, saying if his government wasn't fully legitimate he was in no position to promise independence to the colonies.

Besides all this diplomatic intrigue, Vichy French troops in the Middle East were determined to salvage France's military reputation and honour and fought very determinedly against a British-Free French expedition in Syria and Lebanon, causing big problems and considerable embarrassment for the Allies.

There are a few chapters dedicated to all this in James Barr's A Line in the Sand which is a great read.