Alright, here they are, both senarios:
Neo-Byzantium:
Story: After the War to End All Wars, the Partitioning of the Ottoman Empire began. The Allies agreed to take controll over former territories of the Ottoman Empire: Britain, Palestine and Mesopotamia became her vassals, France, Syria and Lebanon became vassals.
Greece, however, emerging victorious from the Second Balkan War, and with land promised from the First World War, choose now to grab the land they desire, mainly, Constantinople. So they strike, and in doing so, spark the Turkish War of Independence. The Allied Powers, alarmed by this outbreak of war, choose not to get involved directly, instead sending supplies and vehicles (Mainly from Germany) to the Greeks. These supplies helped the Greeks make a steady advance. However, the Turks decide to lead a counter offensive, and as a result, Mustafa Kemal gets killed leading the counter offensive. This is a descisive blow to the Turkish cause, and they are soon crushed under the Greeks newly aquired A7V, while only eight initial tanks are given to Greece, many more are soon in production and aid the cause, the Turkish Independence movement is borken.
And so, the long process of intergrating the Muslim people into an Orthordox nation, and the new begining of an Empire long lost. King Constantine I of Greece is crowns Constantine XII, Emperor of Romaoi. Tensions with the French rise, as 'raiders' raid Byzantine border towns, and then retreat back into French controled Western Turkey (An effort to appease the population). It's the middle of the Roaring Twenties, and every other country has thrown down the wartime spirit, choosing to embrace peace. However, the Byzantines continue to develope military technology, working on a new, better design on their tanks, preparing for a war that looms over the horizion.
Agent of Byzantium:
In a world where Mouamet converts to Christianity on a trading mission to Syria, becomes a monk and later archbishop of New Carthage, and after his death, is recongized as the Patron Saint of Changes.
Thus, the Byzantine Empire is able to reconquer Ispania and Italy(and installing their own Bishop in Rome), as well as the Southern Portion of Frankia. Their rival remains Persia, just as England remains Anglo-Saxon. The Orthodox Chruch thrives in Byzantium, just as a form of Catholicism exists in Northern Europe.
Leaders:
Byzantium: Nikephoros III
Anglia: Osway II
Alans: Goarios III
Thoes are the only mentioned, for the rest, you can either go with 'Leader of...' or a historical leader, just make sure he'she isn't related to Islam in any way.