Yes, I am sure there have been multiple threads regarding Vichy France becoming an active participate in the Second World War on the side of the Axis; this is NOT what I am proposing. I also offer two PODs.
1] - Charles De Gaulle had gone to Bordeaux on June 16th, following the "Union" fiasco that quickly resulted in Reynaud's fall. when they (De Gaulle and several other senior leaders) had learned Petain had effectively come to power and pushed Paul Reynaud aside, they more or less rebelled and decided to flee back to England, where they would soon set up "Free France". Some time during this trip, the aircraft they were in managed at one point to almost get shot down by the Luftwaffe; unfortunately, I do not have the specifics. The fact of the matter is, Charles De Gaulle is killed.
2] - Following the Establishment of the Petain Cabinet on the 15th of June, orders are relayed to the "Marine Nationale" to make sail to North Africa. The British had already at this point made threats to blockade and bombard French ports if an armistice were agreed to in the first place as early as the 31st of May, along with a perceived threat by the French Fleet if it were to be captured by the Germans. Under the auspices of engaging the German and Italian fleets, they would make for the French North Africa after leaving the English ports.
The Armistice is signed just as it was at Compiegne, with no significant changes; France has bowed out of the war. However, the great majority of the French Navy is now in North Africa, with the largest concentration still at Mers Al-Kabir. Operation Catapult still occurs, but the French vessels had properly prepared themselves once the ultimatum had originally been delivered, but the Bretagne manages to still be destroyed (was in the first salvo, remarkably, given the range). Attacks are made on the French Fleet elsewhere, and a retaliatory strike is made on the British base at Gibraltar; however there are now calls for war from the Vichy French Cabinet.
3] - Vichy France had failed to go to war with the United Kingdom by a single vote in the Cabinet in OTL. Here, they receive that additional vote.
While they are now at war with the United Kingdom, Petain makes it clear that they are not to enter into an Alliance with Germany or Italy, and will be fighting the British independently. This is where the other questions begin to come into play. First of all, the United Kingdom now has to divert at the very least some resources to fight Vichy France, even if the war is very low-key.
Adolf Hitler is going to try and get them into the Axis, despite Petain's opinion of which he is ignorant of at the beginning. He may loosen the restraints upon the French military if he figures it may give the British more trouble and hasten their inevitable (in his view) collapse. He may also pressure Japan to NOT occupy French Indochina, which would in turn prevent or at least push back the Oil Embargo imposed by the United States; lots of butterflies there.
All this aside, I have two questions. With these PODs, what would a war between Vichy France and the United Kingdom look like, and how would it affect the Second World War?