Mustafa Ataturk had died only the year before the start of WWII, and Kemalism was very much still alive and well during and long after the war. Germany and Italy intervened in an existing Spanish Civil War, and Hungary represented a country with long and strong ties to the German people. Iraq was basically in the orbit of the British, and Iran was quickly occupied by both the Soviets and the British. So both of these countries had legitimate beefs against the Allies.
None of this was true for Turkey. The Ottomans let themselves be dragged into WWI by the Central Powers and lost their empire over it. OTOH, the founder of modern Turkey commanded at the Dardanelles, giving the Allies their worst strategic drubbing outside of Russia in the entire war. This, coupled with Ataturk's winning of the Turkish War of Independence and his successful radical rebuilding, liberalizing, and modernization of Turkish life meant that the Turks came out of the early 20s with a modest sense of satisfaction.
In short, the Turks in 1939-1941 had no real bones to pick with their neighbors. And after the horrors of WWI, Turkey was one country determined to avoid their past mistakes. The republicans were not going to follow the path of the sultans. By joining the Axis, they had everything to lose and nothing to gain.
And whatever problems the Arabs had with Jewish immigration to Palestine, and British occupation, it was nothing compared to what for them would be the ultimate nightmare of a Turkish army looking to restore their empire, and probably looking for major payback too. Against the Arabs.
Even if there are promises of limited Turkish involvement, not sending any troops south (only against the USSR), politics is about perception. The Armenian and Ionian Greek genocides were not very long ago at this point in time, and the Arabs could remember that the Turks made for harsh taskmasters even to their own co-religionists. This saves a lot of internal security headaches for the Soviets and the British. For that matter, you could expect a national uprising IN SUPPORT of the Soviets within Armenia, typically the quietest of all Soviet republics, after the Russians liberated them from the horrors of Turkish occupation during the Russian Civil War. If the Turks ever made it to Armenia, the fighting would be house-to-house. And the Turks had shown that between their paucity in equipment and poor tactics, they were not nearly as effective in attack as they were in defense.
IF Turkish troops are seen invading Syria and Iraq (1) you could expect to see (possibly) a total reversal on the facts on the ground for the "Arab Street". Even troops raised, as they were in WWI, to act as at least defensive troops to fight against the Turks.
1) Iran really isn't in a position to be invaded by either the Germans OR Turks or both. The terrain in SW Turkey in WWII was horrific for a modern army. They'd have to go through the Southern Caucasus region first, trying to convert the rails to European gauge. And all this from a single rail line leading all the way back to Belgrade before the railroads start to branch out at last. Logistics-logistics-logistics.
MIND, this is not an absolute. I have always believed that just as if Britain were on the brink of total collapse Franco's Spain would enter the war, so too the Turks would DoW the Allies if Russia were about to collapse (say if Leningrad, Stalingrad, Moscow and Astrakhan have fallen). To save themselves from an apparently victorious Axis. Assuming that this is pre-Pearl Harbor, Turkey represents Germany's doorway to Asia. Post-Pearl Harbor, I don't want to guess what happens with Turkey..