Sorry.  I seem to have bogged down on this one.  I’m not sure why.  I do have a few miscellaneous thoughts to share on the scenario though. 

One problem the French would have faced if they had decided to fight on in North Africa is that French-made weapons systems such as tanks and aircraft would have only been useful as long as the spare parts held out.  With no new supplies of parts and replacements from France, the French would be faced with cannibalizing an ever decreasing supply of French-built equipment to keep a few tanks and planes functioning even if the Germans or Italians didn’t launch a major attack.  Ammunition for French-built artillery and small-arms would become a problem if there was a major attack, or eventually even if there wasn’t.  The French would have to use some ammunition for training if that training was going to be effective.

In order to remain militarily viable in North Africa, France would have to get weapons from some other industrial country.  Britain wouldn’t be an option in summer 1940 because it didn’t have enough of anything weapons-wise to equip its own forces, much less the French.  The US was an option for some weapons.  The French had considerable numbers of both fighters and bombers on order from the US, and the US was releasing some warplanes directly from the US airforce to France.  Most of those planes probably wouldn’t have arrived until some time in 1941, but US was still delivering the last few of an order of several hundred Curtis Hawk fighters, along with some bombers and trainers.  Any survivors of the US planes that were actually in French service would also be helpful because the US could give an ongoing supply of spare parts.  The US could also supply considerable amounts of World War I surplus small arms and some artillery—historically the US intended to supply the Brits and French with over 500,000 small arms, primarily Lee-Enfield rifles.  Most of those went to Britain historically, but if the French had still been fighting in North Africa chances are that many of those weapons would have ended up re-equipping evacuated French divisions.

One thing the US couldn’t supply many of in mid-1940 was tanks.  The US had very close to no modern tanks in June 1940.  They had produced about 15 M2 medium tanks (a distant ancestor of the Sherman with a 37mm main gun and lots of machine guns pointing in various directions) in late 1939.  They also had a few dozen machine-gun armed early model M2 light tanks.  There were several hundred World War I era tanks (an improved US version of the Renault FT) in storage.  Historically a couple hundred of those ended up in Canada where they were used for training.  In this scenario I suppose that some of them could have ended up in North Africa.  The US army had just started to get a few light M2A4 tanks out of an initial order of 329 (increased to 365) off of assembly lines in April 1940.  Historically production of the 365 M2A4 wasn’t completed until March 1941.  It might have been possible to increase M2A4 production somewhat, but anything ordered in June 1940 probably wouldn’t have made it off the assembly lines until November or December 1940 at the earliest.  The French might have ended up getting a few M2A1s mediums (slightly improved M2) starting in November 1940.  The US produced 94 of them in late 1940 through August 1941 (out of an order for 126—the rest were apparently cancelled because considerably better tanks were in the pipeline).

Of course the real question in all of this is when the Germans would actually go after French holdouts in North Africa.  The Germans would have had a problem in that the kind of resources they would be short of for North Africa (planes and shipping) were the same things they needed to have any chance of making Sea Lion work or even of convincing the British that they had such a chance.  Weather would have made a channel crossing almost impossible by sometime in October, so the Germans really just had between June and October to do whatever they were going to do against the British.  If they went after French North Africa first they would make an already tight schedule very close to impossible.  My guess is that crunch time in North Africa would come after Sea Lion became impossible.  Both British and German power would shift south in October/November 1940.