For the Soviets to annex Constantinople or parts of Turkey requires a weaker Turkish nationalist movement and a stronger Turkish Communist Party. Ataturk was pretty effective in uniting the Turks against the Allies, and he additionally prevented the Turkish Communist Party from becoming a significant force by means of persecution and consolidation. All Communist Parties are persecuted before taking power, but for them to take power requires weakness on the part of the previous regime. I think the best means for a Communist Revolution to succeed is to have Ataturk meet some accident, and for the nationalist movement to collapse into infighting following his death. The Sultanate would thus be the surviving government of a rump Turkey. The Sultanate would uphold the Treaty of Sevres upholding the partition of Anatolia, and Turkey would become a client state of the Entente, with Constantinople a demilitarized international zone. The Sultanate would be unpopular and vulnerable to revolutionary agitation, which the Soviets would no doubt attempt to exploit. The Sultanate, being subject to the Treaty of Sevres, and the creation of Wilsonian Armenia, would not negotiate the Treaty of Kars, and thus the Soviets would likely seek more territory in Anatolia. The Soviets would additionally seek to destabilize the Sultanate.
Suppose butterflies outside Anatolia are limited and World War II happens roughly in the same way. The Axis invasion of Greece would necessitate the invasion of Greek held Smyrna and Thrace (annexed by Greece under the Treaty of Sevres) in order to completely destroy the Greek government. I suspect Bulgaria would try to lay claim to parts of Thrace. The Axis would then attack Constantinople in order to prevent the British from using it as a naval base and to gain access to the Black Sea. (The Turkish military is a non-factor under the Treaty of Sevres.) I think the Nazi occupation of Constantinople would either provoke the Soviets into entering the war earlier in order to protect the Black Sea, or at least for them to be more alert and prepared for a Nazi invasion. The Sultanate would become a Nazi puppet government, but it would lose authority in mainland Anatolia. This would create the potential for a revolution in Anatolia, and the Soviets would likely occupy Anatolia a la the occupation of Iran to shore up the revolutionary government and to prevent more of Turkey from falling into Nazi hands. There would be a "Battle of the Black Sea" in 1941-1942 as the Soviets desperately seek to capture Constantinople in order to secure a supply line from the British. If the Axis wins control of the Black Sea, I think the Soviets would lose the war- it depends on what's going on in the Eastern Front. Suppose the Soviets win- they would then be able to establish a Communist client state in Turkey.
I don't think Communist Turkey would be annexed by the Soviets- it would be a Warsaw Pact state. Russia has no historical claim to Constantinople, only a religious claim- and the Soviet Union is officially atheist. Unlike Konigsberg, there's no Axis-sympathetic ethnic population for the Soviets to expel. Keeping Turkey communist requires at least nominal Turkish independence. The Soviets could swallow a stupid pill and annex Constantinople as per the terms of the OP in 1945, but I would think Turkey would not stay Soviet very long, given historical Turkish-Russian rivalries. The Greeks, British, and Americans would try to stir up some sort of trouble.