I think it is less a question of what "would" have happened, and more a question of what "could" have happened.
Turkey as an active Axis partner brings some very real potential. Whether that potential would have been realized is a different question altogether.
As I understand it (welcome corrections here) Turkey in 1939 had what would best be described as a WW1 army. A body of infantry, and the ability to call up more infantry, but no real motorization, tanks, or air power worthy of consideration by WW2 standards. We're talking boots on the ground, with rifles, machine guns, artillery, and horses to pull it all.
The German-led Axis activities in 1939 – 1941 were not short of infantry. The German-led Axis activities in 1944-45 were VERY short of infantry, but honestly who cares -- by that time the game was up, and everyone knew it but Hitler. In 1942-43 more infantry might have helped, if it was usefully skilled and led, and most importantly in the right places at the right times. This is where I think the difference could have been made.
What was always missing was any coherent plan by Germany to bring their European Axis allies up to higher levels of capability. Germany quite simply didn't have enough German resources to supply their Axis allies to a useful level, and didn't have the inclination / skills / orientation to orchestrate multi-national industrial cooperation to rationalize their allies' resources effectively so that Italian, French, Austrian, Czech, Dutch and Romanian industry could bring the entire Axis block up.
So to prevent this what-if from becoming too great a flight of fancy, I will assume that the Turks don't magically get all the stuff in 1941 that they didn't have in 1939.
Still … there were several places and times when a half-dozen to a dozen infantry divisions, even if on foot and of indifferent quality, might have made a big difference. Could the Turks have made a difference? I think they could. Would the Turks have made a difference? I doubt it.
A few corps of Turkish infantry in the Levant, and more critically perhaps in Iraq and Persia, could have made a substantial difference.
The British in Egypt would have been greatly distracted from the western dessert campaign, which was repeatedly a near-run thing until late 1942. If they had diverted more forces to try to prevent the Turks from marching on the Suez Canal from the north, they might not have succeeded in preventing Rommel from marching on the Suez Canal from the west. So I can see a realistic scenario where the Suez Canal is lost. And maybe where the British presence in North Africa is lost. That's a game-changer.
More significantly, the Allied effort of opening up Persia as a supply route to the Soviets was very much a shoestring operation, and a Turkish field army of 50,000 or 60,000 foot infantry marching into Persia might easily have turned that whole process upside-down. No Persian supply route probably means the Germans get to at least cut-off, if not actually lay their own hands on, Soviet petroleum by the end of 1942. That turns the battles of the Eastern Front of 1943 and 1944 updside-down, and so turn the entire sequence of events of the land war in Europe upside-down.
Now, is there any reason to believe that a declaration of war by the Turks would have led to this kind of campaigning by the Turks? I can see it as possible, but don't know if or why it would be likely. They might have just sent a corps of men into the Barbarossa campaign (as the Italians did), which would have just meant more bodies to be buried somewhere on the steppes. Or they might have satisfied themselves with some activity in the Balkans, which was such a mess to start with that more stirring of the pot does not produce a noteworthy change, as far as I could see.
One possible interesting additional side-effect might have been shifting German strategic vs. tactical decision-making. Hitler frequently over-ruled his Generals based on his political concerns, and I believe that getting more allies to his side (most important among them being the Turks) was top of his list. He might have been less close-minded to military priorities in 1941 and 1942 if the Turks were already in the bucket.
Might have been. Would have been? Who the h3ll knows.
At least that's my read.
-Mark
(aka: Mk 1)