As a junior partner, no, most certainly not. But you forget historical context. A good percentage of the French forces that fought for the allies in the main line army level later in WW2 where originally Vichy troops that switched sides when the Americans and British beat the Germans and Italians off the continent of Africa.
Taken in the context of this timeline laid out before us, what do the Vichy in North Africa see. At the end of 1940, early into 1941 most would have thought that the British would have been defeated, or at least been put on the defensive. Now here we have a scenario where, except for the Vichy and perhaps some token German and Italian diplomatic and avisory personnel, the Vichy French forces are the only real forces left the the British would have to worry about in Africa.
The British/Commonwealth forces can now rightfully say that they do have what it takes to beat the Germans. They have destroyed the entire Italian army in Africa, the small German army sent to reinforce the Italians, and the German airborne army thrown against Crete. Not war winning accomplishments, but the British have definitely drawn a line in the sand.
While the Vichy French may harbor some grudges against the British for some of the things the British have done towards them, would they realistically want to risk their lives for their wounded pride?