Now that we’re talking about it. I bet a Russian move into the middle east would have put incredible pressure on Turkey. And I’d think it would be pressure to side with the Axis, as both an expanding Germany and Russia would be on both sides.
I think the Turks would negotiate an amicable fold and allow axis troop movements through turkey, and open the Dardanelles to axis fleet movements.
UK would have had serious problems after that, and worse problems - if Turkey was receptive to say… inheriting Cyprus and some other possessions.
At that point it’s even possible that Spain may then have considered the Gibraltar grab more reasonably.