I don't think much changes. The British troops which conquered the Vichy colonies in Syria and Lebanon would instead face the Turks. Syria and Lebanon would stay neutral as Vichy territory. Turkish troops would then have to advance into Iraq to be of any assistance to Italian troops in Africa. I have been to that part of the world. Not easy country to fight in , or even move through. So, let's say the turks do this anyway, I'm not sure why, Iraq is not a strategic objective in WWII. The British forces which historically conquered Syria and Lebanon can now be added to the forces that historically put down the pro Axis Iraq coup and face the Turks. IF things get dicey for the Brits they have some excellent troops in India who can be rushed to the front in Iraq. (Since this is 1940, the Japanese are not a threat). I think the Turks get routed. Now Hitler has to bail out both the Turks and the Italians. Again, this is 1940/1941. Assuming that Mussolini still starts his stupid Greece campaign, the Germans roll over the Balkans and Turkey just becomes an extended front to that campaign. The Germans then move to take out Iraq and join their allies in Syria and Lebanon for a move on the Suez. The Suez falls, Egypt falls, Iraq falls, and Persia falls. The Germans can now take the caucasian oil fields in reverse from Turkey and Persia. The Persians probably make a deal and become Axis minor powers. Iraq definitely does this.

What does Russia do without the caucasian oil fields?

I think this was the most important mistake that Hitler made in the war, not making sure that the Turks would enter the war, or at least allow German troops passage through their country. With the Wermacht camped on the Turks European border as a result of the Balkan campaign the issue could have been forced.