If you have a WWI POD then you could make the Ottoman Empire a little more successful, and keep the Kurdish pieces of Syria (not sure what vilayet this was) and Iraq (the Mosul vilayet) in the Ottoman Empire (which will still become Turkey), then you would unify the vast majority of the Kurdish people in one country. This will massively aid in creating a unified Kurdish liberation movement.
With the vast majority of the Kurdish population now under the rule of the Kurds, I think that the Kurds will probably be able to cobble together a coalition resistance movement. This alt-Turkey will not significantly change history in the first half of the 20th Century, it will however be an important change in the aftermath of WWII. Barzani begins to build a resistance movement through the 1940's, and begins to gain Soviet support. Once the Turkish communists are crushed in the late 40's, Soviet support will switch fully to the Kurdish Liberation Organization (or whatever Barzani's organization is called).
I don't think that this will come to anything. The Turks have American support, the Iranians have American support, the Soviets know that Turkey is not going to be disemboweled by a Kurdish resistance movement. Basically, the Turks can carry out a counter-insurgency campaign and wipe out the KLO.
The only way I can see some kind of successful Kurdish resistance movement is if the Kurds can rise during WWII and get the Soviets to invade eastern Anatolia. This might help out the Kurds, but the Soviets withdrew from Azeri-dominated regions of Iran they invaded, and the Soviet invasion would only increase American aid to the Turks.