This is always a fun “what if.” The French could’ve fought on from North Africa. It was entirely do-able from a military standpoint. But not from a political standpoint.
First of all the French has already experienced a loss to Germany in the Franco-Prussian War. The French were demoralized by WWI and the loss of that generation. They figured that it’d be better to quit and the occupation and terms would be similar to the Franco-Prussian War.
Hitler was a bad guy, but Europe was full of bad guys. Stalin, Mussolini, Franco and a whole host of mini-tyrants.
The French also begin to believe the British would fight to the last drop of French blood. This didn’t sit well and the British retreat at Dunkirk also seemed to confirm this. To be fair, the British at that point were more concerned with protecting their own nation.
Hitler had not expected the total collapse of France so fast. The whole world was sort of shocked that they could be defeated so fast and so thoroughly.
The problem is getting the French to North Africa. With the British help this could’ve been done. The French fleet was big enough to probably do this alone. The Italians at this point were HORRIBLY overestimated. Plus you had Franco to think of.
In reality the Italians were so weak they caused more problems than help and Franco’s Spain was such a mess do to the civil war, neither threat was in reality much, but it seemed VERY different at the time.
To add to this, the British were insisting that the French give them a guarantee they would fight on. The British were not gonna move their ships to help the French get to North Africa, if once safe in North Africa, the French just quit anyway.
The French were thinking “If we were defeated, the British don’t stand a chance.”
So the political situation at the time did not come close to reflecting the reality of the military solutions.
Once you get the French to North Africa, and still in the war you have a bit of a change.
First of all this virtually guarantees all of France will be occupied. This isn’t going to make Mussolini happy. He likes Hitler but at a distance. Will Italy act against the Allies if the French government is in North Africa fighting?
That brings up the whole African campaign, which was largely a waste of Germany’s efforts. There was no real interest in North Africa. Germany’s first goal was oil in Baku, then oil in Mosul, Iraq.
So would this have shifted the campaign out East earlier? It might have. If Italy doesn’t act up in North Africa and Albania, Germany doesn’t have to rescue it.
Oddly enough French surrender brought about a tricky situation. Vichy France still controlled the colonies. And Vichy France, weak as it was, was still better than Italy or Spain. Hitler was eager to get Vichy France to be on his side.
So Hitler wanted this, but both Spain and Italy wanted French colonies. And Hitler couldn’t very well expect Vichy France on his side, if he gave away their colonies to Spain (Morocco) and Italy (Tunisia)
The French effectiveness of fighting is also gonna be in question. If France doesn’t surrender how much effort are they gonna put into fighting if all their population is held hostage by Hitler. Probably little effort, so that puts us back to, “Why go to North Africa, why not just surrender and get to keep a third of the country (nominally) from Vichy”
This is why it’s a fun “What if” 'cause there’s so many thing to explore and so many outcomes that were based on the “idea” of what was, instead of the reality of “what is”