Step 1. Printing press invented in late antiquity, after Romans begin using paper (the PoD).
Step 2. Upon the fall of the western Empire, which happens at a similar time to OTL, the Churches largely claim the printing presses. Although in many places the knowledge of how to make them is lost, the ecclesiastic higher ups are not among them, alowing the Church to monopolize the printed word and far outproduce other writers. Knowledge of the presses is closely guarded.
Step 3. Islam is butterflied, but the Arabs do invade the Eastern Roman Empire and Persia. They are less damaging than OTL, and most of them convert to Catholicism, but their invading and carving off significant portions of the ERE and Persia diminishes their dominant power status in the near east, as do those of the Slavs in the Balkans. The Arabs, lacking a common language with which they and their conquered subjects can communicate, and not wanting to ba absorbed by Greek or Persian culture, begin using Latin in an ecclesiastic and diplomatic capacity.
Step 4. Faster and wider dispersion of knowledge favoring the church prevents many major schisms in the church, including the east-west schism and the Reformation.
Step 5. Church bishops begin abusing the printing press around 1,000 A.D., using it to print purely secular texts and sell them for profit. Eventually, one particularly audacious and short sighted Cardinal in Italy sells the design to several different people. Secular printing becomes widespread, but it still is entirely done in Latin, as everyone worthy of being called educated is well versed in Latin.
Step 6. Bombarded by Latin writings, written word fails to develop in the vernacular languages of Europe on account of being choked out by Latin. Greek alone retains a comperable status in Europe, but its range is limited to OTL Greece, Anatolia, Russia (which is more heavily Greek influenced than OTL), and parts of the Balkans. With time, Latin is not only the language of the learned, but the language of cities too, while local dialects are considered rural and undesirable.
Step 7. Much like OTL, Europe evenutally explodes onto the scene as a powerhouse region full of ambitious imperialists. The Americas are conquered more or less as in OTL, to the point where native languages simply aren't viable for practical use. European city culture prevails in the colonies, resulting in colonies where most people speak Latin as a first language. Many African countries, whether colonized or not, begin to adopt Latin out of convenience and because they lack their own written languages. In the parts of Asia that are colonized, Latin becomes the language of commerce, though local languages are not as thuroughly crippled as in other continents, while in the uncolonized parts native languages are retained (partly through isolationism), but still use Latin as the language of diplomacy. By the time nationalism appears, most states are thuroughly Latinized, and only a few extremist regimes attempt to reinstate native languages, and are usually crippled by the attempt to do so.
Final stats: 45% of people speak Latin as a native language, 30% speak it as a secondary language, and another 20% know a few necessary phrases for their careers. The only other languages with noteworthy numbers of native speakers are Greek, Persian, and Chineese, although there are countless local dialects throughout the world.