You still have to carry that oil from Africa to Europe. That means tankers and escorts and the RN in Malta and Gibraltar would hunt them. The German navy (at least OTL) can't really provide the numbers and types of escorts required, and the Italian navy is already stretched with it's escort duties;
I took that into account before writing Post 86.
Re tankers
The passage from Algeria to Tunisia is shorter than the passage from Italy to Romania so the Italian tankers could make more trips. Therefore, Italy might not be able to use all of French North Africa's production, but it would have got more oil from there than it did from Romania. Also Italy was cut off from its supply of Romanian oil between November 1940 and April 1941 due to being at war with Greece. Being at war with Greece for six months doesn't cut Italy off from its supply of French North African oil.
Furthermore, losses of tankers carrying fuel to Libya are likely to be reduced ITTL so the carrying capacity of Italy's tanker fleet may grow over time. (See below.) Plus Italy might be able to build more tankers 1940-42 ITTL.
There's also the French shipping in the Mediterranean which one of the British official histories said was abundant. The Axis would be able to use the French tankers in the Mediterranean and it's likely that the French merchant marine would have had more tankers ITTL.
Re Escorts
Unless Spain does enter the war the number of escorts Germany could have sent to the Mediterranean is nil.
The RM OTOH had 126 surface torpedo craft in June 1940 and added another 21 by September 1943 plus the 29 corvettes that it completed between June 1940 and September 1943. Losses were admittedly heavy IOTL. In spite of that it suggests that the RM wasn't exactly short of convoy escorts.
AIUI one of the problems with supplying Libya was the capacity of the ports which forced the Italians to send a large number of small and weakly defended convoys. Algeria and Tunisia were more populous countries than Libya with bigger capacity ports so it would have been possible to have a smaller number of more strongly defended convoys.
Also when losses to the convoys to Libya became intolerable the RM ran
"battleship convoys" in which the supply ships were escorted by the RM's entire battlefleet. That's something that it could do to protect the tanker traffic from French North Africa and something that it might need to do to protect said traffic form attacks by Force H.
Another problem the RM had IOTL was that as the war progressed the oil shortage prevented it from attacking the Malta convoys in full strength. E.g. there was only enough fuel to attack the August 1942 convoy with one battleship or two at a pinch when six battleships were available. That wouldn't be a problem ITTL. Plus standing orders prevented it from giving battle with the enemy unless it possessed overwhelming strength, something that the oil shortage prevented it from having. ITTL it had more oil so more battleship convoys to Libya & from French North Africa and more opportunities to engage the RN in overwhelming strength.
The tankers taking the oil from French North Africa could hug the coast (and therefore remain within Vichy territorial waters) under escort of the Vichy Navy before turning north at Tunis and coming under the escort of the RM for the run to the Tyrrhenian Sea. Would the British respect Vichy neutrality in this case? There were plenty of occasions when Britain didn't IOTL. However, this occasion might be the one that makes Vichy join the Axis.
Re The British
The British will want to do everything in their power to stop this. What was within their power?
You mentioned Malta as a base of operations. I think an island in the Mediterranean's central basin is unsuitable for attacking shipping in the western basin. It's on the wrong side of the Sicilian Narrows and the British found it hard to adequately supply it as a base to attack the Axis convoys to Libya, let alone attack the Axis tanker traffic in the Western Mediterranean. Plus Malta might fall ITTL (sooner rather than later) because the RM has the fuel to attack the supply convoys in overwhelming strength.
There will be a limit to the number of aircraft, submarines and surface warships that the British can deploy from Gibraltar. That's a function of the physical size of the place and the quantity of supplies that Britain would be able to send.
The Gibraltar to Malta convoys were attacked by Axis aircraft in Sardinia & Sicily IOTL and so will any sorties by Force H against the French North African tanker traffic ITTL. Said tanker traffic would also have been protected by an Italian battlefleet invigorated with Tunisian oil. (See below.)
What's more likely is that the British will do more to help the pro-De Gaulle factions in French North Africa to rebel and join Free France. It will also do all it can to sabotage the North African oil industry, e.g. blowing up the pipelines or at least try to blow them up. French North Africa will also be occupied at the earliest possible opportunity, but that might not be until November 1942.
involving the Spanish or Vichy for these duties would end up committing them to war vs the allies...
That's Hitler and Mussolini's wet dream of wet dreams!
If Spain's in the war Gibraltar immediately becomes untenable as a base because the harbour and airfield would be under constant bombardment from Spanish artillery. It would take longer for Gibraltar to be starved into surrender. According to the British official history it was decided in late 1940 to make the colony capable of withstanding a siege of 6 months. This suggests that Gibraltar was incapable of holding out for as long as 6 months in the summer and autumn of 1940. Therefore, the Mediterranean's western basin rapidly becomes an Axis lake. The Spanish Navy would not have been of any help, because it only had about a dozen destroyers and IIRC were having badly needed refits in June 1940. However, their help wouldn't be needed.
It also means no more convoys from Gibraltar to Malta. Therefore, all the supplies will have to come from Alexandria and there were periods IOTL when that was impossible. E.g. August 1942. Plus running convoys from Alexandria and Gibraltar to Malta forced the Italians to divide their forces so making it harder to attack either of them in overwhelming strength. ITTL the RM could concentrate on the Alexandria to Malta convoys and in even greater strength than OTL because no ships will be left behind for want of oil.
According to the British official history Vichy joining the Axis is not as ASB as people may think, because it says that it was touch-and-go whether Vichy would declare war on Great Britain or not in the week after Mers El Kébir. Yes most of the French Empire is seized by Free France with British assistance. However, a lot of the French Empire joined Free France before Operation Torch and most of them wouldn't have been of use to the Axis in any case. The important ones are French North Africa and Dakar which are the ones that Vichy is most likely to hold onto.
Vichy joining the Axis solves the (exaggerated) escort problem at a stroke because the French warships in the Western Mediterranean become available. Plus one of the things said by gainsayers in Vichy joins the Axis threads is that even if the entire MN remained loyal to Vichy (which admittedly is doubtful) it would be a fleet in being because the Axis couldn't supply it with oil. No problem supplying the MN with oil ITTL. Although some of the ships outside the Mediterranean would use that fuel to steam straight for the nearest British or Free French controlled port.
