The war would have quickly devolved into a conflict reminiscent of WWI, except with the Germans making further gains than they did in their first outing. The Low Countries would have still been occupied as would Denmark,Poland and Norway. Also, the Anschluss and Czechoslovakia would still be in Nazi hands.

France could have brought its fleet to bear against the Italian fleet in the Mediterranean and removed Italy from the war by early 1941. The French, with assistance of the British could have also taken the war to Germany, launching an invasion into the Rhineland and delayed or prevented any plans that Hitler had for the Soviet Union.

A key detail would be that the US probably wouldn’t have seen the need to enter the European theater of the war. Since France hadn’t fallen to the Germans and it could have still provided potent resistance to Germany, the US would have likely concentrated the brunt of its forces on the upcoming conflict with Japan.

What is rarely spoken about is the fact that in our iteration of history, France could have continued to fight for months or perhaps years after the time that it capitulated in June of 1940. There more than 500k worth of troops in Southern France and over 900 aircraft. Had France been competently by its civilian leadership, these forces could have continued to fight in France or escaped to Algeria and continued the war from there.

Given the Nazis limited ability to project sea power in the Mediterranean there was virtually nothing that they could have done to stop the French had they retrenched to Africa. In fact, France could launched a re-invasion of its homeland as early as 1942-1943 from Algeria,Morocco and Tunisia or even assisted the British in defeating in the Italian Army in Africa far sooner.