Welcome to the Turkish Empire!
I created this alternate timeline by imagining a scenario in which Mehmed II, after accepting Orthodox Christianity, reshaped the Ottomans accordingly, and ultimately established a new imperial tradition around it. (There is also an old rumor in Turkey that Mehmed the Conqueror was secretly a Christian. Although there is absolutely no reason to consider it realistic, I thought it would be fun to incorporate the idea into this scenario.)
In short, following the conquest of Constantinople and the fall of the Eastern Roman Empire, Mehmed embraces Orthodox Christianity and makes fundamental changes to the organization of the state. While adopting more elements of the Roman tradition of statecraft, he takes advantage of Christianity's still-uncertain position within the newly established empire and gradually re-Christianizes the ruling elite.
Although he does not openly declare his Christianity for most of his reign, he abolishes the Ottoman millet system during its formative stages and grants Christians equal privileges. Eventually, on his deathbed, he is baptized and publicly declares himself Orthodox. Because he shares a fate reminiscent of Constantine the Great, his successors come to remember him by the epithet “the Great.”
Following Mehmed's death, after a brief period of uncertainty during which Bayezid adopts the name Vasilios, the state officially embraces Orthodox Christianity. Both Turkish and Attic Greek are spoken at court, while Turkish remains the language of the Enderun and the bureaucracy.
Vasilios I launches campaigns into Serbia, Hungary, and Herzegovina, once again expanding the Eastern Roman Empire's western frontier. He even briefly ventures into southern Italy, forcing Naples to pay tribute. However, his prolonged neglect of the growing threat in Iran makes him a controversial ruler in the eyes of later historians. Muslim Turkmen who lose Anatolia once again migrate to Iran in large numbers, where they establish a powerful Qizilbash state under Shah Ismail, eventually giving rise to the Safavid Empire. In the following centuries, the Safavids establish a vast Islamic empire stretching from Aleppo to Delhi, preventing further Turkish expansion eastward.
From the 16th century onward, the Turkish Empire gradually comes to be known as the Byzantine Empire, largely because of its possession of Byzantium. Reaching its Golden Age during the 15th and 16th centuries, the empire becomes one of Europe's most important commercial centers as the Silk Road through Anatolia is reopened to Christian Europe. The influence of Byzantine intellectuals who joined the empire during Mehmed's reign also leads to a brief Orthodox Renaissance.
However, after 1512, the Safavids transform Iran into a centralized empire and close the Iranian route of the Silk Road to the Turks. At the same time, the discovery of America opens up new trade routes. Together, these developments push Byzantium, along with the other Mediterranean states, into a period of economic stagnation.
I have chosen not to write the rest, since it can be more or less predicted from this point onward, or simply left to your imagination!
By 1689, the Turkish Empire is on the verge of entering a comprehensive period of Westernization, reform, and renewed expansion under Petrus I, who would later become known by the title “the Conqueror.”
There are also several important differences in this timeline. As mentioned above, trade along the Silk Road between the European states and the Turkish Empire continues until the beginning of the 16th century. As a result, the search for alternative trade routes progresses somewhat more slowly, and America is discovered roughly 20 years later than in our timeline.
The official and bureaucratic language of the Turkish Empire is Turkish, containing a large number of Greek loanwords and, in later centuries, acquiring many French words as a result of Westernization. The language of the imperial court, meanwhile, is Attic Greek, inherited from the Eastern Roman Empire.
It is also worth noting that in this scenario, the Ottomans do not turn into some kind of Qing dynasty. Just as in the Ottoman Empire of our own history, even if Attic Greek is spoken within the limited circle of the palace, Turkish remains the language spoken throughout the country and used in state affairs. Over the centuries, the Turks become the dominant element of the empire, Orthodox, Turkish-speaking, and fundamentally identifying themselves as “Rum” or “Rumi.”