Decided to make a scenario of this, based on some ideas I had. I'll post a map soon too.
I didn't do too much extra research into things I didn't already know, but basically we have the Nazis taking more of an interest in the Middle East and the activities of Fritz Grobba and others better funded. Their key message is the same as OTL--anti-colonialist rhetoric against the French and British and antisemitism. Some copies of Mein Kampf in Arabic find their way into the court of Ibn Saud and with the help of Grobba and others, some Saudi princes--including Prince Faisal--become increasingly pro-German. Ibn Saud, having some doubts about the British himself, decides to go along with his pro-German sons and grandsons, perhaps due to his increasingly poor mental health which clouds the mind of a brilliant man once described by European travelers as an Arabian Bismarck or Napoleon. Iraq is the main focus, since they had a larger, better trained army and a much more strategic position, with Saudi Arabia as a secondary partner. 20th century Arab history is not something I'm particularly knowledgable about compared to some members here, but I hope I've constructed something more or less entertaining and decently plausible if you want a "Saudi Arabia joins Axis" scenario.
Ibn Saud begins to condone tribal raids on Transjordan and Kuwait starting in late 1939, after the start of the war, leading to the rapid deterioration of relations between Saudi Arabia and the United Kingdom. In April 1941, the Golden Square coup in Iraq occurs, overthrowing the government of Regent Abdullah and leading to the installation of a pro-German government under Prime Minister Rashid Ali. Through the month of April, the Saudis procede to nationalise Allied interests, confiscate Allied property, and expel their envoys from the country. The Anglo-Iraqi War starts in May, but here the initial Iraqi attack is timed to occur with the largest tribal raids yet, combined with regular forces from Saudi Arabia's military and the arrival of some Luftwaffe squadrons and other German assets. On May 1, Iraqi forces shell RAF Habbaniya, damaging the airbase and some aircraft interfering with RAF operations in Iraq. On May 4, the Iraqis, with some German assistance, take Habbaniya, and soon even more Luftwaffe squadrons are flown in. Throughout May, fierce air battles occur in Iraq and Iran, as the refinery at Abadan is bombed by the Luftwaffe, while no decisive counterattack against Iraq can be assembled due to Saudi harassment of British supply lines as well as the need to protect Transjordan. Paratrooper experience gained by the Luftwaffe is used to great effect during their conquest of Malta in June, albeit at great cost to both the paratroopers and German air assets.
Saudi forces also attack the Trucial States of Abu Dhabi and Qatar in May seeking to gain control over disputed areas and reclaim lands. Kuwait is attacked in June, as Saudi Arabia looks to link up with the Iraqi forces besieging Basra, the main lifeline of British control in the area. They likewise help support the forces of the Imamate of Oman in revolting against the Sultanate of Oman to seek greater autonomy and overthrow any foreign influence in the region. Through the summer and well into the fall, British control in the Middle East is looking shakier and shakier. British forces are increasingly needed in both North Africa and in the Far East, as the Japanese threaten their key holdings there with their attacks on Thailand (and Thailand's subsequent defection to the Japanese side) and then Malaysia.
However, the Germans are unable to exploit the region for a decisive victory. German supplies have to travel a large distance to the front and the losses inflicted to the Luftwaffe are adding up. Further, Hitler is concerned that these supplies are needed more in the Balkans, and even moreso, for the top secret Operation Barbarossa, the invasion of the USSR. Abadan has received additional protection and security, so the offensive against that key refinery is losing steam. The Arab armies are likewise thought of as asking for too much in terms of supplies and weapons they need to continue the fight. Iran, a potential ally, has been pushed into further neutrality thanks to the constant violations of their airspace in the conflict and most importantly, a number of British warships, aircraft, and soldiers that appear ready to pounce on the country if needed. And the attack on Pearl Harbor by Japan in December led to war with the United States, and regionally, provoked Ibn Saud to nationalise Aramco and kidnap several Americans for ransom. Infuriated by this act, and with the press filled with lurid tales of Christians and Jews converted to Islam at gunpoint and the murder of missionaries (both true and exaggerated), the United States issued an ultimatum to both Saudi Arabia and Iraq, demanding compensation for the attacks against Americans and American interests (which had been occurring since 1939) as well as a demand to hold a peace conference with the British Empire to return to the status quo ante bellum. Both nations refused, and thus the United States joined the war against the Arab Axis [1], with Congress unanimously passing an official declaration of war on the Kingdom of Saudi Arabia and the Kingdom of Iraq in January 1942.
In winter of 1942, the Arab Axis is at their height, with the Trucial States, Kuwait, Qatar, and the majority of Transjordan conquered. Iraq and Saudi Arabia share the occupation of Kuwait, Transjordan, and Palestine, planning to divide the spoils at a later date. Attacks on Palestine are occurring, but fiercely opposed by the British, loyal Arabs, and Jewish militias. The initial gains in the south of Iraq have been reversed, with Basra only holding out thanks to the Royal Navy ensuring a clear supply route. In Kurdistan and North Iraq, the Arab Axis is slowly pushing back the Allied forces. Atrocities are common--several pograms have occurred against Jewish and Christian communities in the region, while captured soldiers are routinely mistreated if not outright murdered. The worst offenders are the tribal levies of the Saudis--with little discipline in their ranks and extreme fanaticism, they kill and loot at will, and have outright murdered hundreds of POWs, including Major General Bill Slim, captured during an attempted breakout from the Siege of Basra.
The global circumstances were changing by spring of 1942 however. The Germans recalled the majority of their forces from Iraq to the Western Desert for the attack on Egypt. British air superiority began to show for the first time, as the Arab armies began to melt back. Further, the British chose to move several more brigades to the theater, seeking to end the conflict as fast as possible. The British also secured the allegiance of the Imam of Yemen, Imam Yahya, in exchange for territories in southern Arabia. Yahya, fearful of Saudi attacks against the once disputed city of Hodeida, had been building up his military with British support and preparing for the worst the past several months. Yemeni forces invade Saudi Arabia in late April 1942, backed by the British.
The Arab Axis themselves began to suffer from internal disputes between both Saudi Arabia and Iraq, as well as inside each country. The Iraqis despised the Saudis as violent fanatics more concerned with imposing their ideology than strategic concerns, while the Saudis viewed the Iraqis as munafiqun and even polytheists. The Saudis alienated other allies as well, by not restraining raids against the Ibadis of the Imamate of Oman. Their relationship with the Germans, while excellent at the highest levels (although Ibn Saud, increasingly ill, is curiously absent from many meetings), is poor at the level of the tribal sheikhs, who view the Germans as a corrupting influence and little better than the British. Allied propaganda toward these sheikhs, as well as Arabs at large, focuses on Nazi hostility toward religion, promotion of secularism and atheism, their imperialist intentions toward the Middle East, and allegations of Nazism being linked to paganism. This propaganda, appealing to the many Arabs who had rejected fascism, causes great internal dissent in Iraq and Saudi Arabia as tribal levies begin to melt away and soldiers defect.
In April, British forces landed at Dubai, Muscat, and Hodeida, and with local assistance, pushed inland against the Saudis. Similarly, new offensives occured in the north against Saudi and Iraqi forces as the supplies from the Germans dry up and internal disputes sap morale at every level. Pushed back on every front, the Arab Axis goes for a scorched Earth campaign. The only bright spot is the successful use of tribal levies to slow down the Allied advance. Flooding in Iraq likewise hinders the advance of the British trapped in Basra, although they have already begun their breakout, while decisive victory cannot be achieved thanks to the need for additional forces in North Africa, considered a more important front.
Summer sees the Saudis reorient their forces to focusing on the Yemeni invasion, while pulling out of Oman entirely. A further blow to the Arab Axis comes in June 1942, as the Germans choose to end their support for the Arab Axis in favour of focusing further on the USSR in their 1942 offensives--only a token force of observers is left, and supply shipments almost entirely cease. The British forces expel the Arab Axis from Transjordan by the end of July 1942, although the territory falls into civil war between Arab nationalists (with their Wahhabist Allies) and more conservative pro-Hashemite factions. The decisive breakout from Basra occurs in August, as the Arab Axis is pushed back from the city. Attacks from Transjordan, Yemen, Oman, and Egypt strike the Arab Axis on all sides. Meanwhile, persuaded by the British and Soviets at the threat of invasion, Iran joins the Allies and invades Iraq in August 1942. American Lend-Lease aid pours into Yemen, Oman, and Iran helping to equip local forces and relieve strain on the pre-existing British supply lines.
The decisive moment for the defeat of the Arab Axis comes at the Battle of Fallujah in November 1942. Seeking to cross the Euphrates and take Baghdad, British forces battle a large number of Iraqis and Saudis, as well as the remnants of the German forces in the area. The remainder of German, Iraqi, and Saudi air power is thrown into the campaign. In the end, they are unable to prevent a crossing of the river, and tens of thousands of soldiers are captured. In December 1942, a battalion of US Marines lands at Dhahran in Saudi Arabia, the site of a massacre of American oil workers at the behest of the Saudis. The Marines secure much of the coast, the main oil producing area, and provide security to rebuild the oil wells to help supply local forces. They will not push inland due to logistical difficulties, but establish a working relation with Arab tribes in the area and beyond, beginning a period of internal conflict in Saudi Arabia.
1943 is the final defeat of the Arab Axis. January sees Yemeni forces enter Mecca, clearing out the Saudis and their allies--in one of the most famous pictures of the war, often seen accompanying descriptions of the Middle East theater, Yemeni forces raise the flag of the Arab Revolt on a building overlooking the Kaaba. Meanwhile, Afrika Korps being pushed back leads to more British forces pouring into Arabia and the Hejaz, accompanied by local tribal leaders and the Hashemites. In February, the Saudis are defeated at Medina, and the ousted regent of Iraq, Prince Abdullah, proclaims the restoration of the Kingdom of the Hejaz as its king. By the end of February, few Saudi forces are left in the Hejaz.
The loss of the holy cities of Islam cripples Saudi legitimacy, and leads to their final defeat. In March 1943, a revolt long in the making erupts in the Nejd--the heartland of the Saudis--under tribes associated with the historic Ikhwan Revolt in the late 20s, including the Otaibah, the Mutayr, and the Ajman, who had suffered reprisals from Ibn Saud for their participation in the revolt. Supported by British (and American) arms, they condemn the Saudis as munafiqun for the murder and theft from their ancestors while embarking on similar campaigns only a decade later, condemn their association with the atheists, polytheists, and idolators in Germany and Italy, and demand their removal. They create an army sometimes called the Second Ikhwan, which wages war against tribes still loyal to the Saudis. The majority of remaining Saudi soldiers revolt or defect over this, with the message of the Second Ikhwan appealing to them. The Trucial States (including Qatar), and Kuwait are liberated in this time, while the Imamate of Oman declares war on the Saudis.
In May, almost 2 years after the start of the war in the Middle East, Ibn Saud is assassinated by the Second Ikhwan, and Crown Prince Saud takes the throne. Upon suspecting his younger brother, former crown prince Faisal, is supporting the Allies, he has him killed along with several "disloyal" Saudi princes. To outsiders, this internal conflict in the House of Saud showed the decadence of their family. The Al ash-Sheikh, descendents of Muhammad ibn Abd al-Wahhab, condemned the Saudis soon after, accusing them of numerous sins and demanding their abdication in favour of a proper Islamic ruler--Muhammad ibn Ibrahim Al ash-Sheikh led the condemnation of the Saudis. With the ulema turned against them, the defeat of the Saudis was near.
Meanwhile in Iraq, Baghdad was encircled on every side by Allied tribal forces, the British, and the Iranians. Rashid Ali surrendered on June 1, 1943, in exchange for allowing his supporters to go into exile. With Iraq lost and defeats mounting against the Axis globally, King Saud of Saudi Arabia ordered his forces to surrender on June 6, 1943 as Second Ikhwan forces approached Riyadh. Some Saudi fanatics would continue the fight in the deserts until early 1944, when those sheikhs were killed or captured and their forces dispersed.
The Allied occupation of Saudi Arabia was a mix of local, British, and American forces. The Kingdom of Saudi Arabia persisted under King Faisal, who took control not long after the Saudi surrender. Faisal had briefly flirted with a pro-German stance, but in the end had abandoned it out of opposition to his brother and rival Saud. However, the Saudis had become associated with corruption and violence in the eyes of both the world at large and the Arab world. Faisal abdicated in 1946, ending the Saudi state in favour of the new Kingdom of the Nejd, under the leader of the Second Ikhwan [2] and his shura council of allied sheikhs. The Kingdom of the Nejd would be recognised as the successor state to Saudi Arabia, while recognising the independent of the Hejaz under Abdullah of the Hejaz, ceding the northern regions to the new loyalist regime of Iraq under the young Faisal II [3] as well as Transjordan under Abdullah I, and giving large swathes of territory to Kuwait, Qatar, Yemen, Abu Dhabi, and Oman. The gains of Ibn Saud over decades of skilled campaigns and shrewd diplomacy had been reversed overnight. These successor states engage in a purge of Saudi loyalists, destroying the majority of opposition as well as imprisoning or executing many of the perpetrators of war crimes, although some would end up rehabilitated and wind up in powerful positions in the successor states of Saudi Arabia.
In the end, the events in the Middle East would not change the course of the war much. Historians debate the severity of the effects of the attacks on Abadan as well as the number of soldiers being tied down in Iraq and whether they contributed to British failures in 1941 and 1942, or whether the German commitment to the campaign in the Middle East represented an overextension of German resources during their decisive showdown with the Soviet Union. When the war ended in April 1945, with Hitler's suicide, Allied forces meeting the Soviets at Fulda, and the overthrow of Hirohito and his militarists by a clique of Imperial princes and IJA/IJN leaders in the Kyujo Incident, the USSR came out on top to start the coming Cold War, with their "iron curtain" stretching from Trieste and the Tirol to Lübeck in the north [4].
[1] - A complete hodgepodge of Arab nationalists and traditionalist regimes united based on opposition to France and especially the United Kingdom and antisemitism. TTL, their leadership includes figures like Ibn Saud (and some of his family), Amin al-Hussein (Grand Mufti of Jerusalem), Rashid Ali, but they are unable to gain the support of the majority of Arabs and the Middle East which leads to deep divisions in those countries. If not for more pressing matters, it's doubtful their war would have lasted as long as it did.
[2] - I don't know who would be particularly plausible as a replacement for the Saudis in this circumstance--it could well be the Saudis, but maybe giving the leadership of the Nejd to another powerful tribe (like the Otaibah, Mutayr, etc.) might be plausible too.
[3] - The Iraqi military is so wrecked by the war and the "reconstruction" of Iraq so thorough that decolonisation is much simpler in Iraq and the Hashemites survive to this day, with the coups against them defeated.
[4] - I can't write a WWII scenario without a bit describing the outcome. Basically the Soviets do better due to butterflies, leading to a Bavarian Republic (including the Tirol/South Tirol) in a situation similar to Austria, an Austrian Socialist Republic (like East Germany), Trieste being joined to Yugoslavia, and a larger East Germany (including bits of Lower Saxony like Brunswick and the city of Lübeck). On the other hand, the American occupation in Japan allows for the Japanese under the young Emperor Akihito (installed by his uncles with his great uncle Naruhiko as Regent)--eventually the longest reigning monarch in history--to keep Chishima and Karafuto Prefecture due to concerns about the coming Cold War. Japan still lost Taiwan and their Pacific holdings, as well as Manchuria and Korea.