I agree, a lot would have to change with Turkey's external siuation or its internal politics.Nations don't go to war without a reason. What reason would Turkey have to attack the Soviet Union during WWII?
Getting back tot the logisitics question. If this railroad terminates at Erzum, either stops there, or it is impractical to make effective use of a Soviet rail connection then it will be difficult to sustain offensive operations into the mountains. Even light infantry divisions will require 200 - 250 tons per day. Much more if there is intense fighting requiring large ammounts of artillery ammunition. Aside from the ammunition there would be fuel for vehicals & fodder for draught animals. A army of ten divisions with corps & army support groups will need over 4,000 tons per day for a sustained moderate level of attacking. Add in a mechanized corps, a cluster of forward airfields, ect.. & the need can run up past 6,000 - 7,000 - 8,000 tons per day. Assuming the railroad to Erzum can carry that consistently taking the supplies from the railhead to the combat battalions will be difficult over the roads like that in the photograph posted above. Would the Turks have the motor vehicals to carry the cargo, or would they depend largely on horse & oxen dragging wagons?
I supose the potiential choke point of the railroad ferry at the Bosphorus could be gotten around by bringing the cargo via barge on the Danube River & hence to the docks on the east shore. Barge might be more efficient than the railroads through the Balkans for much of the bulk cargo.