I don't see the French seizing oil fields as realistic. First, if they try through Syria to intervene in Iraq they will face the British Indian Army and Commonwealth troops, both of which could have outfought the demoralized Vichy. The Australians will recognize how dangerous this could be and will send additional troops beyond what they've already sent. In 1941, when the Brits in North Africa were facing huge problems, they were still able to lead an Allied coalition to take the French Levant in June-July without great difficulty. If the French had declared war on the Brits a year earlier, Churchill who in OTL forced Wavell to undertake the Levant expedition or else, would have instantly seen that having a Vichy force in the rear of his North African forces and bordering on Iraq was simply unacceptable. Jan Smuts of South Africa, whose influence in the Mideast theater was enormous, also would have seen this and backed Churchill to the hilt. The East African campaign against the Italians would have been postponed except for holding actions in Sudan, and Vichy Syria would have been seized with dispatch. THEN the Italians in East Africa would have been crushed, maybe by some of the blooded troops of the Levant campaign.
I don't think the French had the forces or the morale to take Cyprus. The premise of this thread is that they would not ally themselves with the Italians or the Germans, at least not at first. But if, say, they only ally with the Italians, it wouldn't be enough. Only the Germans could take Cyprus and they could only do so with paratroopers. But look at Cyprus on the map--it's an island too far for German paratroopers. And it simply wasn't in Hitler's line of vision in 1940--he was concentrating on the BoB and already planning to conquer the USSR, thus getting all the oil he might need.
Once the Vichy regime is out of Syria, the Brits only have to deal with the French in Algeria, Tunisia and Morocco. If the French are not allied with the Italians, then Malta retains its importance. If the French don't attack Malta, which in effect would place them in alliance with the Italians, then the seige of Malta unfolds pretty much as in OTL. But maybe there's a butterfly effect: the Brits (a) stay out of Greece, (b) the Germans never get a chance to get rolling in Libya, and/or (c) the war with France and the potential for a French-Italian alliance forces the Brits to take Malta a lot more seriously and send much more food, oil, planes, antiaircraft, etc. much earlier before the siege really tightens.
As to the idea of no DeGaulle, no Free French--or no Mandel, no Free French--I don't believe it. Someone else would have risen to the challenge. Someone who in our timeline was possibly a quite obscure resistance person. The need for such a patriotic figure was so great and so obvious, a leader would have emerged. Would have HAD to emerge. Saint-Exupery, perhaps? So there would have been some kind of FF force in the Levant campaign, either larger or smaller than the one IOTL.
I stick to my guns; Vichy, after expressing its rage over Operation Catapult and getting trounced as a result--and after getting a very strong message from the U.S. Ambassador--and after a few easy pickings among its colonies overseas are occupied--will get the message. Continue this stuff and you lose all your colonies. Continue this stuff and we'll send the whole lot of you to the guillotine after the war. So the French "war" against the British turns into a phony war of posturing and posing.