A very interesting what if regarding WW is the continuation of hostilities by France after the German invasion of 1940, and by all accounts, this was nuch a far-fetched possibility : a group of influent Frenchmen (among then former Prime Minister Daladier) fled France on board SS Massilia for such en eventuality :
From Wikipedia :
In June 1940 she carried a large number of prominent politicians, including 27 members of the French Parliament, who fled from Metropolitan France to French North Africa after France capitulated to Germany and Italy.[12] The group included Édouard Daladier, Pierre Mendès France, Georges Mandel, Cesar Campinchi, and Jean Zay[13][14].
But would the Allies' cause have benefitted from such an occurrence ? The answer, in my opinion, is not as obvious as it seems.
A fighting France would have brought 2 things to the common struggle : its non-negligible colonial empire, and a powerful and modern navy indeed.
The Navy would have been a welcome addition in the Mediterranean, especially the 2 modern Dunkerque class battlecruisers, and, shortly afterwards, the battleship Richelieu which was (slightly) better – in my opinion – that anything that would have opposed it, with the Jean Bart coming much later. Add a few modern light cruisers, the powerful heavy cruiser Algérie, a score of super destroyers (Fantasque, Vauquelin, Audacieux classes) and many other valuable escorts and destroyers, there is no question the naval balance in the Mediterranean would have been tipped outright. Okay. But for how long ? There was no integration between the Marine Nationale & the Royal Navy, ammunition, spare parts would have become soon a problem. The fact is that even for the mighty US industry, refurbishing the French ships after 1943 was of questionable value, despite the insistance of the Gaullists. I wonder how could Britain have sustained it ? Especially since most ships were suited for the Med, with « short legs » making them hardly suitable for escort duties across teh Atlantic.
Now, on to the Empire. The raw materials provided would have been a boon indeed, but there was no real shortage in Britain ; Shipping tonnage was the choke point, with the U-Waffe taking a big toll on it (and a lot of French ships had been seized outright during Catapult). TheEmpire could have provided a lot of recruits, as happened in 1943. But most of these were native troops, well commanded & courageous, but ill equiped and trained to fight German forces. The French Army was not ready to move on Libya, and had a defensive posture there (the Mareth Line).
But the main problem, I think, would have been the huge change in the strategical situation, which, in my opinion, would have brought a change of priorities in the German Command. Operation Barbarossa was possible because, by all means, in mid 1941, Hitler could consider Britain as contained. The French Empire in the game changed a lot of things, making it necessary to protect Italy and « the soft belly of Europe ». This is where Operation Felix comes into play. It never took place, but it was a contingency plan to invade Spain. The Spanish army, il equipped after the civil War, would have been at a loss to stop 2 German Army Corps rushing through all the way to Gibraltar. It is likely also that Franco, placed in front of the alternatives, would have yielded and joined the Axis (as far as october 1941, the option was not cancelled by the Wehrmacht).
en.wikipedia.org/wiki/Operation_Felix
In both cases, the fall of Gibraltar, the capture of French Morroco or at any rate, Tangiers, would have closed the Western access to the Mediterranean and created a catastrophic situation for Britain. So, even though I truly despise Pétain, and his collaborationist policies, in the end, I think the continuation of the war by the French Empire would have caused more problems than benefits. But, of course, your mileage may vary.