Maybe. That Soviet invasion might make the Turks even more pro-Western and considered more European by the Entente so that it's not as economically and politically snubbed as it has been.
However, the Soviets'd most likely have a field day pushing through Bulgaria and crushing Turkish Thrace PDQ despite their woeful issues with logistics and lack of small-unit doctrine that made the Finnish campaign such a bloody mess. They won't have the winter slowing them down and can bring their tanks and artillery to bear where in Finland, those advantages meant very little.
The Soviets could bog down in several places, if the Turks fortify the right places and have adequate air support, but I don't see them being able to hold them out of Istanbul for the six months it'd take the British and French to land enough troops, gear, and ammo to make much of a difference.
The colonial units handy in Iraq, Egypt, Syria, and Algeria won't have the gear or training to fight mechanized troops with artillery, tanks, and aircraft bombing and strafing their positions.
The butterflies of Entente intervention could really screw the USSR short and long-term. Anyone for a replay of the Crimean War?