My best guess is that this requires a huge multinational empire - such as a Carolingian empire that holds together.
If people are speaking lots of different languages in the Empire, the language of administration probably stays Latin, and that may well be THE language of communication and learning.
OTL, you had lots of nation-states developing, which will naturally tend to use their own (or at least the ruling class's) vernacular for administration. This gets written languages in vernaculars wide spread, and you need to prevent that for your challenge.
Note that French wasn't recognized as anything other than 'bad Latin' until the Oath of Strassbourg 842 when Charlemagne's empire was divided into a 'French' bit and a 'German' bit (to massively oversimplify).
Keep that empire together, with no 'national' subunits, and Latin stays the only recognized language for longer.
If you also do what the Romans did and the Chinese did, moving officials and army units from one part of the empire to another, then that, too, will push a unified Latin usage. If you are a merchant in Paris, and you need to address an official who was born in Vienna to get your licences to trade, you need to conduct the conversation in Latin.
And so on.
While peasant dialects will develop, they might never get written. Look at Plattdeutsch or Afrikaans which didn't get regularized written forms until long after the languages were developed.