With a PoD before 1500, change history in such a manner that various vernaculars (whether Romance, Germanic or Slavic) never rise to prominance and Classical Latin (even it's somewhat simplified form of Medieval Latin) remains the sole high language of Europe and later, after the Industrial Revolution, is adopted as a spoken language by the wider masses. Something close to the rise of Hebrew on steroids.
Depending on what is meant by "the sole high language" there are a few possibilities.
Perhaps if a major criminal case occurred in the very early Roman Republic where two officials were caught conspiring with each other via messages written in Greek.
The Senatorial response to that includes the passing of a law declaring it illegal within Rome to write in a non-Latin language. Speaking and reading other languages was perfectly acceptable, and it was still possible to write in other languages as long as it is done outside Rome.
Over time, it would have been considered suspicious and distasteful to possess anything written in another language.
Fewer and fewer educated Romans would have the ability to read Greek. Many Greek texts would be translated into Latin, and then the original work is destroyed. These trends would all feed into each other; as less Greek texts exist there is a lower incentive to learn to read them, as less people can read them the stronger the distrust for Greek writings.
Eventually someone would formalise what writing is permitted, resulting in a series of strict grammar and orthography rules for writing. Any changes in spoken Latin (at least among those capable of reading) is going to be slower as a result.
While never an official policy, as Rome expands it very slowly purges all other forms of writing from the territories under it's control. Traditionally, to be Roman (ie to be civilized) is to only write using (Classical) Latin. Most people would be illiterate, but of the Romans capable of reading and writing the vast majority of them would not be able to read other languages.
There would be times where the written languages of conquered peoples are subject to more or to less tolerance, greater or lesser suppression. The gradual erosion of other written languages makes the spoken form of Latin relatively stronger, and result in a larger percentage of Europe converting into Latin speakers (compared to OTL).
Alt-Christianity (or whatever religion ends up dominating the Roman Empire) would have to adopt the same belief in order to convert the literate Roman population. It would be considered heresy to not use Classical Latin for writing.
Written Latin would remain static due to the combination of a formal set of rules and the religious requirement. The spoken languages would divide after the collapse of the Roman Empire (Vulgar Latin evolving into Romance languages, various tribal languages continuing to be spoken, etc) but all writing would be identical throughout Europe.
Members of the Church, some aristocrats, and some international merchants would continue to speak Classical Latin. Most people however would just speak their local tongue even if they can read; a King in Norway and a King in Italy might not be able to speak with each other, but would still be sending letters in a language understood by both.
During the Industrial Revolution increasing literacy rates means that more and more people can read Latin, and with that comes a growth in the number of Latin speakers. Mass-media (newspapers, followed by radio, and followed by television) would use Latin to maximise the number of potential readers/listeners/viewers. By the time colour television is invented spoken Latin has become the single most widely used language.
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