Your point is partially correct. Both sides in Baku will have supply problems.
The Germans have supply problems before they ever get near Baku.
The difference is what Germans have mostly a steppe with developed rail and road network at their back, plus they are sitting right on oil wells.
There are no oil wells in the east Ukraine/lower Volga region. And if there were, it wouldn't do the Germans any good for a long time. Here's the clue:
oil wells don't gush motor fuel. Gasoline (or diesel) has to be refined from crude oil. That requires the right industrial facilities (in good working order). Iran (right now) doesn't have the necessary plant and has to import gasoline.
There are oil wells in the Cis-Caucasus (at Maikop and Grozny), which would fall to the Germans, but would the Germans capture any functioning oil wells? The Soviets would probably wreck them. The Germans could eventually get them producing again, but only after a lot of work requiring equipment hauled all the way from Germany.
British on the other hand would have to ferry oil
gasoline
2,000 km away is Egypt and British India. Iran, where the Allies have massive supply operations, is 400 km from Baku.
As about British starting from the control of Baku after Soviet Union collapse, it is highly improbable.
Soviet Union leaders, officers and general populace deeply distrusted Allies. SU can withdraw occupation forces from Iran if situation on Caucasus would become dire, right. But chances for British coming to Azerbaijan without confronting Red Army (or its remnants) are nil. The memories of Entente intervention in the Russian Civil War were too fresh. The most possible is the formation of the "Free Russian Army" modelled after "Free French Army", but much less controllable because of the distrust i mentioned earlier. In particular, that army would definitely refuse to take any assignments outside Soviet Union, and would not allow any Allied troops to enter former Soviet Union territories.
The premise of the thread is "USSR falls". Stalin is probably dead or in prison, and very probably whatever Vichyoid rump government that is left in unoccupied Russia and has capitulated to Germany has repudiated Communism.
German troops (the
enemy) are advancing on the Caucasus from the NW. Are the ex-Soviet forces going to fight their Allies, right up to when the Germans shoot them in the back? I don't think so.
IOTL, Germans advance failed while within artillery range of the westernmost oil wells.
The Germans captured the westernmost oil field at Maikop in August 1942, and held it till January 1943. During the intervening months, the Germans tried to get some of the wells back into production, and failed almost completely. They obtained only a few thousand barrels.
P.S. Your main misconception is the too narrow definition of "Baku oil". Baku was the heart and transport hub of oil-bearing region, but the wells of the region were scattered several hundreds kilometers around.
There is a field at Maikop (800 km WNW of Baku), and a field at Grozny (500 km NW), but nearly all the other fields are in the immediate vicinity of Baku or in the Azerbaijan plain to the SW, within 150 km.