There would most definitely be some development of Algerian oil by France. While short term there is the need for capitalization developing the oil fields creates employment and not make work employment which is vital. Money paid by French consumers, including the government, to the French national company for Algerian oil is not resulting in a flow of hard currency to the USA or the foreign companies that control Arabian and Persian oil. This money now recirculates in France. Once Vichy surrenders, the oil will be part of reparations for Germany and Italy Doesn't help Japan (prior to PH) as it is too far away and no way to get it to Japan (not enough tankers).
The oil would potentially help Franco, however the food issue still remains and that would take several years even if the Ukraine is conquered and "pacified" to get harvests back up to prewar levels. Unless the Axis is doing better militarily Franco stays out.
I expect the situation between the UK and France is going to get even stickier. The RN is going to be going after tankers bringing oil from Algerian to Europe, no matter what flag (and the USA won't be shipping this oil), and the RAF may attempt to bomb the oil fields, although range and accuracy are going to be against them (not sure if there are any potential airfields existing or that can be built rapidly close enough for UK bombers in 1940-41). Needless to say the RAF bombing the oil fields and the RN sinking French merchant tankers is not going to make Vichy, or Pierre in the street, have a warm fuzzy about the UK on top of Mers-el-Kebir.