The thing is, its basically impossible not to have a Free France.... and there are inevitably going to be certain colonies which have to defect to the British, because their position is untenable otherwise...
I don't know that that is inevitable. There were French colonies in what should have been "untenable" positions which remained loyal to Vichy until after TORCH: the Caribbean colonies, St. Pierre and Miquelon... Somaliland, Réunion, Madagascar, and Indochina don't seem very tenable.
The only French colonies which are certainly "untenable" for Vichy are the French posts in India, and that's not enough to launch "Free France". Nor, IMO are New Caledonia and French Polynesia.
However, there are a lot of French soldiers and sailors in British territory - some of the troops evacuated from Dunkirk, the crews of interned warships. Britain may try to recruit some of these for a "Free French Army", with minimal success I'd think. So no "Free France" there either.
What could happen later on: OTL, in 1942 a secret faction formed in French North Africa. The goal of this group was to seize power in FNA at the time of Allied invasion, so there would be no resistance, and then be recognized by the Allies as the government of France (since Algeria was legally part of France), with lots of political and financial benefits. This failed: the group was not strong enough to gain complete control, and the Allies recognized Darlan instead. and De Gaulle insisted on Free France sharing power after Darlan was assassinated.
ATL, Britain and then the US will be much more interested in such groups or plots. Supposing TORCH roughly as OTL, that FNA faction would have undisputed Allied support, with no conflicting claims from "Free France".
OTOH, Communist efforts to organize and lead resistance in mainland France will also have less competition. The "France" recognized by the Allies in FNA will have no presence in mainland France, and will be playing catch-up in this area. The Communists may try to exclude the FNA group from any participation as mere ex-Vichyite opportunists. Post-liberation, there could be open conflict, similar to Greece (though probably not as flagrant). This could have consequences for post-war French political life.
Last edited: