Frankly, I don't see it. The Greek Navy operated with 5 old french-built submarines and 2 italian-built destroyers from Alexandria for the duration of the war. The greek boats had pretty good availability. The French operate their own ships, they have shops and base personel in Mers-el Gebir and half a dozen other bases and possibly a lot of the Brest and Toulon personel got away as well.
What sorts of shipyard facilities did France have in Algeria?
Anyway, any and all British facilities are available, and any shortfall the French have in personnel or machinery to upgrade their own facilities are again supplemented by British help; surely the British would quite like a major shipyard facility or three on the African coast of the Med! (Which by the way makes me wonder, what kind of shipyards might exist already in Egypt, associated with the Suez canal perhaps?)
Once the USA enters the war openly, French operations in the Caribbean region can be supported by US ports in the southeast; in addition to New Orleans, there's whatever is at Corpus Christi, Galveston, Biloxi, Mobile, Pensacola, Tampa, Key West, Miami, Savannah, and Charleston before we get to the major facilities on the Cheseapeake at Norfolk and Newport News. Then of course the major ports of the US industrial northeast and on to Halifax--I just want to focus on what is in theater, and we are talking west Atlantic operations already. And of course beyond CONUS ports, which are the most secure and best supplied to be sure, aside from whatever Britain maintains in the Caribbean there is the US Panama Canal Zone; surely the USN has major facilities there too. And in this day and age, well shoot I forgot Puerto Rico, and was going to mention Cuban harbors too.
Too bad none of that (except maybe Havana, though I don't suppose the Cuban government will be persuaded to risk a DOW by the Germans nor do I suppose Cuban facilities are all that capable) is available yet, but anyway whatever the British have to support their Caribbean operations normally are on hand. French Guiana is there, but not developed to any great degree. I wonder now if France fighting on might influence some Latin American nations to come in openly on the Allied side even before the USA does, which might offer the French and RN ports in Brazil or Argentina--Brazil would generally be the most useful, between French and British possessions in West Africa and the northeast bulge of Brazil, they could make it very very hot for U-boats trying to sortie into (or return home from) the South Atlantic and beyond. Sadly both Brazil and Argentina had pre-war contacts with the Germans and movements and leaders flirting with fascism. But I do wonder what the state of relations was between Paris and these nations, especially Brazil. I would fear the northern Brazilian ports aren't much better facilities wise than any of the Guiana ports (British Guiana, Dutch Surinam) that are available already. But anyway it would be nice to set up bases there.
I would wonder if either Britain or France would consider the idea of purchasing blimps from Goodyear, but I suppose not as the US is cagey with helium to lift them. (OTL, wartime produced Goodyear blimps designed for inflation with helium have in fact been operated by German operators who substituted hydrogen for helium, and there was never a fire. But of course the Germans are shooting back at any blimps they see (not actually their general inclination but they were ordered to do so OTL and did so) and would be more likely to consider trying to shoot down a blimp cost-effective if they knew it was lifted with hydrogen and therefore more vulnerable than was the case OTL when the USN operated them all.
Anyway if they could have the blimps and Brazilian bases, they are pretty wonderful for detecting U-boats--or even more so, scaring the subs into diving and trying to hide, which immobilizes them and neutralizes their threat pretty effectively, just by being seen. (Diving and hiding is the U-boat captain's impulse and instinct, knowing that even if they can send the blimp down in flames in very short order with their deck guns, they have sent up a flare telling the Allied combined ASW forces just where to look for them. It doesn't matter whether a blimp can sink a sub by itself or not; what matters is finding the sub. A blimp and even its crew is a good tradeoff for neutralizing a U-boat).
OTL once the USA was in the war and Goodyear production lines were ramped up to capacity, and meanwhile Brazil was persuaded to join as a nominal ally and thus permit USN operations, blimps were in fact based there, as well as at British bases in the Caribbean, and from Brazil a blimp had range to ferry itself all the way to West Africa, which is how a number of them wound up in the Mediterranean, doing minesweeping as well as ASW work.
So that's my excuse to ask these things. Realistically even if the US government were open handed with helium, and/or the British or French were willing to cross their fingers and operate with hydrogen instead (extra risky as the Goodyear designs assumed helium and thus did not take the same sorts of design precautions they would if planning to inflate with hydrogen, but as noted safe operation can be done, at least if one is not shot at too much--furthermore shooting even a hydrogen inflated airship is less likely to set it afire than one might think, most WWI Zeppelins shot down came down because their hydrogen leaked away and they had insufficient lift, not because they caught fire--sometimes they did and that was spectacular, but less often than one might guess. Anyway, it would take a while for Goodyear to ramp up production, and a while for French or British operators to gain proficiency and invent the tricks USN operators did over the course of the war OTL; by the time a serious number of purchased blimps are in operation in the appropriate theaters, the US is likely to have entered and to largely take over these operations complete with helium in trainload lots.
However if even a small scale practical operation hunting U-boats is going the day the USA does enter, perhaps it would save time, accelerating the USN operations getting up to speed. Certainly if Goodyear is already placing orders for dozens of blimps for France or Britain, production for the USN should start rolling out faster.
) or it might still be the destroyers just because they are in effect Monopoly money just as I suspect they mainly were OTL--it looked better before the War Cabinet and the virtual Parliament of British public opinion that Britain got "something," never mind what, and pawning off the old destroyers, when FDR had every intention of getting the construction of new shiny modern ones approved in gross lots as fast as he could get it through Congress, was cheap for the US. But in fact both FDR and Churchill would be well satisfied to have the USA simply gifted the bases I suppose. Quid pro quo looks better though, and made it easier for the USA to claim to still be neutral.