This really has too many other variables to tell. What of Wilsonian Armenia? does it also survive? Same as the international zone of the straits; is it demilitarized, or does it become a militarized border? Does Greece have Ionia and Eastern Thrace sans Constantinople, or does it only have the latter?
If Greece only maintains Thrace, then it's doubtful anything would change. If Ionia exists, and the Armenians have nominal control over Wilsonian Armenia, as well as French Syria still maintaining Alexandretta... Then there's quite a bit of temptation.
However, if they entered, it would not be at the fall of France. It would be after the Italian invasion of Greece, and the subsequent German reinforcement. This Fascist Turkey would join in publicly for the spoils in Greece (revanchist claims), but also to intervene against the USSR (and, if they are still independent, Armenia/Georgia/Azerbaijan, depending on butterflies). In order to accomplish this, I would imagine that there would have to have been some buildup over time (basically, a visible transition, a military buildup), and the British and French may or may not have responded accordingly. The Arab states would certainly have noticed; they would not want to be reincorporated into the New Ottoman Empire, which is what it would have been, so support for the Axis might have fallen in the Middle East in response.
Note that a revanchist Turkey likely would not give up its claims over Mosul as well, pushing Iraq closer to the British. Same if they plan to annex parts of Syria; hostility would bring opposition.
In the end, the amount that Turkey would contribute industrially would have been minimal; the biggest boon would be the amount of resources they could provide and putting more bodies against the Allies in the Middle East and in the USSR. That might mean that the Axis pushes much more quickly towards Baku, making the relocation of resources for the USS far more important (considering the terrible terrain of the Caucasus, though, any fighting would be long and hard. And if we're assuming Wilsonian Armenia, they have a lot farther to go. Baku may or may not fall, but it will require more resources than OTL (meaning Germany likely advances farther elsewhere). Baku is simply too important.
Note that the Turkish were mostly armed with WW1 era munitions during WW2; that would have to be supplied either from Italy or Germany (or captured supplies), so Turkey would likely be getting the short end of the stick: those being supplies that couldn't be spent elsewhere.
End result being that the USSR is bled even more than OTL, Baku likely doesn't fall at a great cost, and French/British/Colonial armies in the Middle East hold off the Turkish advance and push into southern Turkey. French and British likely offer Kurdish independence and recognition in exchange for support, so fighting in Anatolia likely becomes a long, hard slog through the region. The USSR is even more insistent for a second front, so perhaps Italy is more invested, or maybe Normandy is moved up? Even so, the allies move into Europe earlier. (Assuming the USSR is still able to function, then their advance maybe up to 6 months behind schedule. maybe even more).
The Allies move in and move the final boundaries east, with, possibly, an allied-leaning Yugoslavia (Kingdom of?) and a Kingdom of Greece restored. Greece will probably want the straits after such a war (and to make a land connection with Ionia), the Kurds get their independence, the USSR likely moves in and takes northeastern Turkey (see USSR claims), and the USSR fumes about having been bled by the Germans and not being able to get their pound of flesh otherwise.