Militarily:
France won't completely fall until late July 1940 or early August 1940, this is both for logistical reasons as the Germans need to resupply and because resisting in the Rhone valley on on river lines like the Loire or he Dordogne is a lot easier.
A French led attack on Italian North Africa is a given, but when it can take place is a matter of debate and discussion. I personnaly say that it can be launched as early as July, taking into account the large supply bases and forces available Algeria and Tunisia.
Longer tem the Pacific theatre will be a lot easier for the allies. Singapore fall's might be avoided and Sumatra might be held onto by the allies but at a cost. In my opinion this could open cans of worm in the long run. The priorities of the Americans and Anglo-French will be different. The Americans will want to strike Japan at the jugular and help China. But the Anglo-French will aim to reconquer their colonies at the expense of helping China and the unrelible Chiang.
Germany could delay Barbarossa until 1942 or strike in 1941 anways but with reduced forces and a consequent "drive to Moscow".
The war will overall be shorter by a few months in my opinion.
Diplomatic:
The Franco-British alliance is intact and the "their finnest hour" will be a shared experience. The Americans will have to fit in previously established command structures and such, they will gain command by numerical weight eventually, but the allied war effort won't be dominated by America to the extent it was OTL.
French Empire:
For a few years,
the French colonial Empire will become France herself. This will have interesting consequences post war and the Algerian war will be butterflyed away. The shared experience of the colons and indigenes will be enough to lower resentment and allow for a peaceful transition in Algeria and elsewhere. The ride may be bumpy at times, but France will leave with her head held high TTL and without having betrayed her principles as was the case OTL.
FFO posits large scale economic development in North Africa. Their case is good in my opinion, but I won't follow that route in Sword of Freedom. The case for heavy industry in Algeria is weak, chiefly because local energy sources are not large enough to justify this. Significant economic development of lighter industries is nevertheless guaranteed to happen in my opinion, coupled with land reform and agricultural modernisation in North Africa. Algeria may not be able to produce steel and guns, but it has the capability to make uniforms, ration packs, some ammunition and to act as a massive resupply base.
Occupied France:
The Laval government will be a Quisling like puppet. Requisitions and privations will be a lot greater than OTL and occur in all areas of France TTL. I can consequently see more civilian deaths due to poor nutrition and cold compared to OTL. The harsher occupation could result in much more post war resentment towards Germany. Especially if scorched earth is adopted towards the end of the conflict, as the allies reconquer France.
The overall bill for rebuilding France will more expensive than OTL as well.
France overall:
The collaboration won't be seen in the same way as it is OTL. It is important to realise than the war is still a complicated and thorny subject in France even nowadays. There is still a lot of shame dating back from the collaboration and the fact that it the then political class condoned Pétain actions. Some even says that in a way the current Franco-German "axis" was created or made possible by wartime collaboration. Certainly, in a lot of circles the war and its consequences dampened any ideas that France could be an independent power on her own, thence Europe.
Gaullism won't emerge TTL and French politics will be drastically different. The communists won't be as influential and the SFIO will be the main leftwards force in French politics post war. No gaullism means that the post war French right won't have a clear leader, but rather a lot of influential personas like Reyanud, Mandel, De Kerillis and a few others.
The third Republci still won't survive the war nevertheless. But TTL fourth Republic will be more parliamentarian than OTL fifth. The French Prime Minister TTL will have roughly the same powers and influence as OTL British Prime Minister.
I followed the english forum for FFO until the team split and still follow APOD (don't have the necessary grasp of french to keep up with the remaining core FFO work). TBH, both have/had their fair share of issues.
A certain subset of the French FFO team saw the work as a chance to "restore French honour", resulting in a few occasions when methedology seemed to degenerate to find notable battle from OTL, cross out OTL result and scrawl "teh Frenhc Winz!!!"; a French colony aquired cutting edge industrial infustrucutre with no rational explination or a break through technology popping up a year early because the French suggest it. Of cause, that's only some small part of the work and there was also some quite brilliantly researched and interesting tales coming out of intense "simulation".
The French team has a lot of research power and a lot of expertise. But yes I do agree that there is a certain bias towards what you mentionned in a way.
My main quibble with them is that sometimes they are way too convergent with OTL, especially with regards to post war outcomes and consequences. Still, I realise that this is not something they are focusing on to a great extent.
Equally so, on the APOD side, Mark seems to be steering more and more towards using APOD to explore his "ideal fantasy world" in which the British Empire endures forever more. That and his rather questionable view that Casha nd Carry followed by Lend-Lease was almost an active conspiracy by the yanks to strip Britain of everything they could (now, I can see that the programmes had negative effects for the UK and I dare say elements in the US government were trying to use it to advantage, but I can't buy his line that everyone from Roosevelt on down was driving it mainly to break Britain)... Still, the rule of cool outbreak with the Great Deeds arc has kept me coming back.
Mark has lost a lot of credibility in my eyes by casually saying things like "The British Empire was a voluntary association of nations" or "A lot of FFO stuff regarding Algeria is projection France current issues on the past". the Japanese mass killer diesease outcome on the Isles he is proposing is also borderline ASB.
I can't deny that his expertise is massive, but I am tempted to say that he has binders sometimes and that he is ideologically motivated. The stuff about Chiang sorting himself and the Nats issues out for example is not plausible.
To be fair, probably every AH has a bias of some sort. I am very much aware of this myself as far as Sword of Freedom is concerned. Thence the current extended research phase. Chiefly because I want to explain why things will happen in the way they will. Moreover, while I am very aware that to an extent Sword of Freedom, will be a Anglo-French wank especialy post war. This wanking will have had a cost, both internal and external.