Get rid of Petain - government was on its way to North Africa but ordered not to! Reynaud lost his nerve; Weygand wanted to stop fighting.
The gold hoard was in Dakar along the Polish one.
Armee de l'Air was evacuating the aircraft that could manage crossing the Med.
How to do it - Petain was 84 let him stay in Madrid as ambassador to Spain. Find somebody else than Weygand to run the Army or perhaps with Petain out of the way he may grit his teeth.
The French wrote that the problem was Reynauds mistress and let her die in a car accident - perhaps without Petain that may not be necessary.
Having goverment leave Bordeaux for North Africa and continue evacuation your are on your way. Question is how many troops may be evacuated - go with the Colonial infantry tankers Polish Czechs and whoever decide to make a run for it. A couple of 100,000? 3-4? The important thing as Churchill saw it was to get Luftwaffe POWs out - some 1000 aircrew that won't fly against Britain or NA this autumm! Get the French Radar developers/operators and code breakers out too.
Marine Nationale is a given it's light forces did make a run for it in 1943 and a number made it. Here more are on station overseas and will survive.
Sitting on the gold hoard the French government is able to keep its procurement in the USA of aircraft going - quite vital as there is no production base in NA to speak of. As French designs wears out the Armee de l'Air will receive its ordered H-75s, P-40s or whatever those will be named in export, P-38s when available, light bombers DB-7 and Marylands, B-24 early version LB-30s, which won't be diverted to RAF use! - then they will go buy ammunition petrol and all the shit to wage war.
No de Gaulle to annoy Churchill and Roosevelt!
With the French still alive and kicking the Japanese won't bully for bases in Indochina - if they do the French may dispatch more troops there.
When the French army get its shit together in NA - where a new army is training to replace the one sent to France following the declaration of war - its bye, bye Benny's Colonial Empire.
No Rommel in NA - no German/Italian reinforcement of Rashid Ali in Irak - no British resources spent neutralizing the Marine Nationale or Operation Exporter or Madagascar or whatever used to neutralize the French OTL. No airbattle against the Vichy over Gibraltar.
Franco even more than OTL shut up and do nothing with the French on his doorstep in NA.
Following clearing of Libya the French armed forces will need another round of reequipment and training before new operations. At least they will need new tanks to replace those lost - OTL the Italians did take out Commonwealth tanks in NA so they will take French ones out here.
The Germans - they need more troops to garrison France. Hitler have still done what he wrote in Mein Kampf - France have been defeated. It will stay for quite some time off Europe. Now what? OTL there was Battle of Britain - will it happen? Or is Hitler going for the French in NA? The Kriegsmarine is in no way able to go to the Med so that option would befall the Italian Navy which may be taken out by joint RN/MN action and that will most likely be high on the agenda in that area.
So it may still be a go at Britain as Churchill will be even more resolute here than OTL in keep fighting. Having lost quite a number of experienced aircrew the Luftwaffe will do worse than OTL over Britain.
Next up a lull in campaining where the Allies will build a strategic reserve in the Middle East which means more troops stay in Australia and more British/French troops available if the Japanese decide to go for the Greater Coprosperity Sphere by force.