Which Turkey? The Ottoman regime in Istanbul or the Kemalist one in Ankara?
When in 1920? Before or after the Treaty of Sevres was signed (August 10, 1920)?
Why does the USSR want to war Turkey?
With these aside, Turkey was absolutely in no position to fight Soviet Russia in 1920, especially not the Sultan's government. The Western Allies had effectively gutted the Ottoman government in Istanbul in March 1920, and Turkey was effectively partitioned as per the Treaty of Sevres later on in the year. And that's not mentioning the fact that Turkish nationalists under the leadership of Kemal were fighting against the Ottoman government -- and Kemal was friendly toward the Soviets as well.
Lastly, the British and French were exhausted after years of war (see World War I), and having failed to topple the Bolsheviks in the intervention of 1918-20, it's unlikely they would participate into yet another war against the Soviets.
So to answer your last question: this is not a realistically possible PoD. You'd need drastically different circumstances for the Turkish-Soviet war to take place in 1920.