As others have noted, it would be extremely unlikely for Turkey to do that. However, as a thought experiment, let's consider the immediate effects of Turkey joining the Axis.
The most effective moment for this would be 6 April 1941, the same day Bulgaria's neutrality ended with German forces attacking Greece and Yugoslavia from Bulgaria. We'll further assume that with Turkey joining the Axis, Germany sees an opportunity to capture the oilfields in northern Iraq, and therefore commits substantial forces to the theater.
(Yes, this is a diversion from BARBAROSSA; but having greatly increased fuel supplies would more than compensate.)
Also, on 3 April, the anti-British "Brotherhood of the Golden Square" seized control of Iraq. OTL, a month later Iraqi forces attacked British bases in Iraq; they received a tiny amount of aid from Germany, by air, with the connivance of the Vichy French colonial authorities in Syria.
Finally, Rommel had attacked in Libya on 24 March, and by 6 April was approaching Tobruk.
One other thing: British forces were in the final stage of the East Africa campaign; they took Addis Ababa on 6 April. Some Italians held out in the northern mountains till September 1941.
IOW, the British were already fighting several fires at once.
Thus, while it is likely that British intelligence would detect German deployments in Turkey, it seems unlikely that the British would have any forces available to deploy in response.
OTL, Britain was defeated in Greece, but held off Rommel in Libya, put down the Iraqi rebellion, and conquered Vichy Syria, while completing the mop-up in East Africa - just barely.
ATL: as with Bulgaria, German troops are deployed to attack immediately. This will be a strain on the Germans, as Turkey's railroads are not robust. But then the British forces have to be supplied by sea from around Africa or from India and Australia.
It should be possible for Germany to send a panzer corps with air support to Iraq, which is more than enough to defeat the British forces there, IMO. The British may be able to hold out in far southern Iraq, around Basra.
French Syria is the next question. It's between Axis Turkey and Allied Palestine. The main rail line to Iraq passes through Syria (from Aleppo to Mosul; though oddly, for about 300 km it runs along the Turkish side of the Turkey-Syria border). There are also oil pipelines running from Iraq to the Mediterranean through Syria. The French there are Axis-sympathetic; as noted, they assisted the Germans in aiding the Iraqi rebellion.
The German position in the area is very strong. But it's one thing to allow a few aircraft to transit through French airfields, or to allow use of railroads or pipelines through French territory. It's very different to allow combat troops to deploy and operate there. That's a step Vichy France will be reluctant to take.
OTL, Britain responded to the much smaller French collaboration by invading Syria. ATL, France has done more, but Britain's position is far weaker.
OYAH, there will be fighting in the desert between Iraq and Trans-Jordan. French Syria and neutral Saudi Arabia lie to the north and south, and the borders are mere map-lines across trackless desert. Both sides will be likely to violate those borders.
Yet another question is Cyprus, which is close to Axis Turkey, meaning a lot of air battles and possibly Axis invasion. (That may depend on whether ITTL Crete is a Pyrrhic victory for Germany as in OTL.)
If the Axis gain the full use of Syria, they will certainly want to take Cyprus and secure the SLoC to Syrian ports, and holding Syria would make that easier. Or if the Axis takes Cyprus, they would certainly want the use of Syrian ports. So getting either means getting both.
Possible longer-term knock-ons:
OTL, Iran was forced to accept British-Soviet occupation in summer 1941, and suspected pro-Axis Shah Reza I was deposed in favor of his son. Iran made no resistance. ITTL, Germany can directly support Iran.
If the Axis gets Syria, ISTM likely that they conquer Palestine. This could be very bad for the Zionist settlers there. OTOH, Hitler might see this as an opportunity to Finally Solve the "Jewish Problem" in a "humane" way by deporting all the Jews there - where most of them would starve to death anyway. (It would be impossible for the 400,000 Jewish settlers in Palestine to provide for 3M-6M destitute and helpless refugees. But that wouldn't be Germany's problem. Their hands would remain "clean".)
Axis Turkey means a fighting front in the Caucasus during BARBAROSSA. The terrain is difficult, but not impassable, and there is a narrow coastal plain along the Black Sea. OTL, the Soviet Black Sea fleet took refuge in Batumi, just north of Turkey; that won't be possible ITTL. The Black Sea could become an Axis lake. The Axis could even bring in major warships of the Italian Navy; it wouldn't take much to wipe out the small, mostly decrepit and obsolete Soviet force. (OTL the Axis had nothing.) With a secure water line of communication, Axis forces might be able to advance from western Georgia to Azerbaijan.
Later on: the Middle East would become a central theater of the war. If Iran followed Turkey into the Axis, or the Allies forced their own entry into Iran, there could be active theaters of combat in Baluchistan, where British India bordered Iran, and in NE Iran, on the border with Soviet Turkmenistan.
The Axis might occupy Egypt and drive the British out of southern Iraq, but it would be impossible for them to march all the way down the Persian Gulf and Red Sea. With the US in the war in say 1942, the Allies would counter-attack. It would be a huge strain, supporting a large army there, but the same would be true for the Axis. By late 1943, the Allies would have cleared the Axis from the Middle East and North Africa. The US and Britain would be preparing to invade Europe from the southeast.
In any case, the outcome will not be good for Turkey, which would be campaigned over and damaged as much as Italy was OTL.
One other important point: can Germany get any oil from Iraq? Probably yes. The Iraq oilfields are much larger than the one Soviet oilfield the Germans occupied, much easier to capture, and much better connected to Europe. How mcuh? And what difference does that make? Idunno.
And besides, all this was a thought experiment off an essentially impossible premise.