Assuming French and British troops remain in Constantinople/Istanbul during the interwar years, how would they react once France surrenders in June 1940? the city provides a valuable base that can be used to bomb oil fields in Romania but it hasn't joined the war just yet.
I mean both Greece and Turkey probably haven’t joined sides yet with both essentially probably hedging their bets. Though once news of the defeat at Dunkirk and France's fall spreads, I'm betting that Turkey would be gunning to join the Axis powers as Greece would be seen as leaning pro-Western Allies. Hitler would be likely to accept such an alliance as he'd want Turkey to help obtain the Russian oil fields in the Caucuses.
Turkey also weakened by the war, might have lost some territory to the soviets who might have been pressing their claims. Either way, if they don't press their claims during the interwar years, Stain is going to probably press it in the aftermath of WW2 where you'd probably see bouts of ethnic cleansing similar to what happened to Lwow in otl where Polish Galicia was ethnically cleansed of its population and East Germany lost its Eastern lands.
Then again, Italy would probably be somewhat hesitant to take on Greece compared to otl, as they might seem a lot stronger than in our timeline. So after the fall of France, I expect Greece to immediately send missives to London warning of an imminent Turkish attack. Greece no doubt would be a theatre of major importance to the allies in Europe.
Once WW2 ends, assuming the Allies re-occupy the straits, it would be interesting to see Istanbul/Constantinople develop as an independent city-state.
I doubt "Istanbul" would end up as an independent city state. Greece has historic claims on it and after shedding blood to help defend against Axis Tyranny there would be a strong case to take the city.
During the aftermath of the British defeat in France, I could see Greece ally closely with Yugoslavia. Greece might just straight up annex the European part of the city as the British evacuate the city, and with their navy, they'd be able to cut off the Turks from the straits.
Though they'd also have the issue of Smyrna as that would face unrest from local turks and the Turkish army trying to take it.