I have some ideas for what the Middle East will look like by the 1960s-1970s.
Turkey:
It has been alluded to that Turkey will experience some form of Super-Islamism. For that reason, I think Turkey's internal politics will resemble the Islamic Republic of Iran. I don't think they will have a single Caliph. Once the Islamists get into a position to overthrow the Sublime Porte, I think they will repudiate the Sultan's claim to being Caliph and do what Ataturk did by transferring the Caliph's power to the Grand Assembly. Unlike in Ataturk's reforms, the Grand Assembly will retain the religious powers of the Caliphate. The Islamic Republic of Turkey would claim that democratic election of the Grand Assembly, which holds the Caliphate's power, constitutes proper Shura. The President of Turkey, like the members of the Grand Assembly, would be democratically elected. The President, though, would be a puppet of the Supreme Leader of the Islamic Revolution, a member of the Grand Assembly who serves as a Prime Minister figure. The Supreme Leader would be able to issue decrees on behalf of the legislature, veto the Grand Assembly's laws, wield the Republic and Caliphate's moral authority, and lead the Turkish armed forces.
The flag of the Islamic Republic of Turkey would be a red and black bicolour, with red toward the hoist and black toward the fly. A thin white stripe would divide the two halves. There would be a white star and crescent on the red side of the flag.
Turkey would be ruled by a single legal party, the National People's Union (symbolized with a red-white-black tricolour). It would be organized along Eduist/Leninist lines as a national vanguard party, but would be a big tent ideologically. There would be a far right wing, the National Bloc, arguing for right-wing populism and a return to the social structures of the Ottoman Empire. Its political colors would be black and white. On the other side would be the People's Bloc, arguing for Islamic socialism, left-wing nationalism, and social democracy with universal charity and welfare. Red and white would be this bloc's colors. I can't see there being any center of which to speak, but the two extreme sides would be able to find common ground in their shared nationalism, populism, and Islamism. Outside the NPU, there would be an Eduist guerilla opposition in eastern Anatolia, especially in Kurdish-majority areas.
Turkey's foreign policy would be a weirdly syncretic mix of Neo-Ottomanism and Pan-Islamism. Turkey's main goal would be to regain control of the Middle East and Balkans. This means isolating and defeating the League of Tsars' member states, Armenia, Israel, and Iraq. Armenia would probably be Turkey's first target of conquest. The Neo-Ottoman foreign policy would also include reclaiming Cyprus and building up a naval presence in the Mediterranean and Black Sea. The Pan-Islamist initiative would mean fueling Islamist movements in the IPRR (Caucasus and Central Asia), Albania, Bosnia, North Africa, West Africa, Zanzibar, Indonesia, and the Indian Subcontinent. Turkey could rein in these movements by creating the Organization of the House of Peace to create a united front against its global adversaries. Turkey's main ally will be Rashidi Arabia and possibly an independent Indonesia, and its main enemies will be Israel, Iran, and the IPRR.
The Turkish military would consist of the Land Forces, Navy, Air Force, and National People's Guard. The Land Forces and Navy would be relatively conventional branches. Most of the Army's equipment would be indigenously produced and based on Swiss and Dutch models. The Navy would consist mostly of cruisers and frigates, with some leftover battleships and one or two archaic aircraft carriers. The Air Force would be unusual in that it is overloaded with archaic, obsolete airships carrying 1940s to early 1950s biplanes that have been retrofitted with jet engines. The Turkish Air Force would also be responsible for the "Wall of Minarets," Turkey's anti-rocket and anti-artillery air defense system patterned after Israel's Iron Dome. The most recognizable symbol of the Turkish military would be the white stahlhelm helmet with a red star and crescent painted on the front.
Syria:
The Syrian Republic will probably begin to exist once the French evacuate the region. It will probably be a constitutionalist state in theory but a military junta in practice due to the political influence of the military leaders that drove out the French. However, a truncated version of the political culture of the French democratic period would remain, as would the Napoleonic Code as a basis for the legal system. The National Bloc (no relation to Turkey's) would be Syria's dominant party, led by Nazim al-Qudsi. Al-Qudsi's administration would be primarily interested in retaining the support of Aleppo's notables and the oil, textile, and mining industries. Most of Europe's cheap clothing and automobile fuel would come from Syria. Another one of these notables, Khalid al-Azm, would be the Prime Minister. By the 1960s, al-Azm would have been a veteran of Syria's local ORRA equivalent.
The Syrian flag will be either the OTL flag of the Arab Kingdom of Syria or that of the Syrian Republic.
Syria's biggest internal problem will probably be the issues of Lebanese and Kurdish secessionism. I think Lebanon will be a nexus of sectarian conflict and internal violence for years to come even ITTL. Beyond Syria's borders, its foreign policy will probably involve supporting the IPRR, Europa, and Iran all at once, but opposing the NUSA, Israel, Turkey, and Rashidi Arabia. Syria's policy toward Turkey will be ambivalent, and swing between mild support and equally mild opposition. I expect the Turkish-Syrian rivalry will mainly express itself through football.
Iraq:
I think that Iraq will remain a kingdom. In reaction to the looming forces of Iran, Syria, and Turkey, Iraq might pursue an alliance with Rashidi Arabia and the Arab Gulf states, and become the center of monarchist Pan-Arabism. Maybe Gamal Abdel Nasser will defect from Egypt to serve as the Iraqi PM or commander of the Iraqi military. One of Iraq's foreign policy goals will be to regain Palestine and Jerusalem from Israel. During the French evacuation from the Levant, Iraq might seize control of Jordan and parts of the West Bank.
Israel:
I think that Israel will become the home of the European and Russian Jews some time in the 1940s or 1950s. However, I think that there will be a series of events leading up to the Gathering of Israel. Firstly, Zionism will develop within the Haskalah (Jewish Enlightenment) tradition. This will create a Jewish reform movement that takes on liberal and proto-Illuminist overtones during the late nineteenth century. Lev Bronstein (Trotsky) will take over the intellectual and political Zionist movement in the 1910s, and begin arguing for "Zionism in one country," because of the irony of it. Then, the Napoleonic Levant receives waves of Jewish immigration. First from Russia under Tsar Victor, then from Germany and the IPRR, then from Germany and Italy. By the 1950s, Palestine will be full of Jewish immigrants. When the Napoleonic Empire can no longer sustain its control over the Middle East, Israel will stage an uprising to free itself. The Israeli forces will seize the Temple Mount and probably destroy the Dome of the Rock and the Muslim areas of Jerusalem. Physically, 1960s-1970s Jerusalem will resemble OTL Tel Aviv.
Israel's flag will be white with a Star of David at the center and a rainbow band through the middle. There will be white space between the rainbow and the blue star.
Israel's political system will resemble that of the IPRR due to the lingering Illuminist influence on Zionism. It will probably feature democratic centralism as an organizing principle, and have Trotsky as its protector until his death at some point in the mid to late 1950s. Even after Trotsky dies, his Israel Party will govern Israel as a one-party state. It will be an intensely militaristic state, and will justify its standing army by using it as a labor army between conflicts. Agriculture will be based on kibbutzim, on which the majority of the non-urban Israelis live. The kibbutz effect will probably kill off Israeli birthrates starting in the late 1970s, causing Israel to require immigration to survive. The Israeli government will probably market immigration to American Jews and advertise travel extensively in the NUSA. Also, I think that Israel will develop a caste system to measure how "Arabized" a person is and restrict the rights of the Palestinians to the same degree as Apartheid or Jim Crow.
Israeli foreign policy will probably feature alliances with the NUSA, Iran, and Pharaohnic Egypt. Although Israel will be a pseudo-Illuminist state, it will be the IPRR's enemy after the 1950s. Once Israel has broken its relations with the IPRR, it will pivot to become the NUSA's "greatest ally." Some enemies besides the Illuminist states will be Syria, Iraq, Turkey, and the Arab Gulf states. Israel will have one of the largest and strongest militaries in the Middle East, including new equipment from the NUSA, legacy gear from the IPRR, and captured Iraqi, Syrian, Turkish, and Rashidi equipment. Captured enemy soldiers will probably be thrown in gulags and used for slave labor.
Egypt:
Taha Hussein will probably be the President of Egypt after its independence from France for a short while. After that, he will be the Pharaoh. Hussein will be a secular dictator preoccupied with Egypt's pre-Islamic past, and try to rebuild the Egyptian state in the image of the Ancient Egyptian civilization. The Arabic language will be turned into Kemetic newspeak, mosques will become temples to the Pharaoh and civic virtue, ancient cities will be bulldozed and rebuilt as neo-Kemetic metropolises, and old literature will be banned. The Egyptian government will be run in a purely despotic way, with the dictatorship enforced by the rabidly nationalist military and the omnipresent State Security Investigations Service.
Egypt's flag will be blue with three white triangles, representing the pyramids, in a horizontal row across the middle.
Egypt's foreign policy will be based on control of the Nile and the Eastern Mediterranean. They will probably get bogged down in a guerilla war in Sudan against Rashidi-supported rebels. Egypt will be allied to Israel and to the NUSA by proxy (without the NUSA's formal acknowledgement of the Kemetic Infie heathens). There might or might not be an Egyptian-Iranian alliance. Turkish-Egyptian naval rivalry will dominate the Eastern Mediterranean, and Egypt might claim Cyprus as a target of expansion. All the Arab states and Turkey will be Egypt's enemies.
Iran:
Some sort of dictatorship by the Shah in the style of the Pahlavi Dynasty or Imperial Japan.