I wonder whether it would have been possible for the Romans to adapt the Greek alphabet to writing Latin, and to then gradually begin writing Latin in the Greek script more and more, for reasons essentially of cultural snobbery. Relatively few Latin-speakers would have been literate, and many of those would be bi-literate, in both Latin and Greek; this would ease the transition. So would the rising Greek influence in the Empire, associated with the capital moving to Constantinople. Gradually, texts that had been written in Latin would tend to be translated into Greek, but even texts that remain transcriptions of Latin language also at least get rewritten in the Greek script; the skill of reading the increasingly peculiar-looking Roman text gets rarer and rarer.
An opportune time for completing this transition and locking it in even for Western Europe would be the time when Christianity is adopted by the Empire as its state religion; the Greek alphabet would have extra cachet, since the Gospels were written in Greek and the Eastern regions of the Empire were the places where conversion to Christianity had made the most progress. There would then be not just cultural fashion but two powerful institutions urging the transition forward--the Roman state, and the Christian church. If the Church has adopted (and adapted, to accommodate more sounds than the base alphabet covers) the Greek script as its standard before the Empire starts to disintegrate, then even the eventual Roman rite of the Christian church will develop scripts based on Greek rather than Latin script. That would make it universal (in, to be sure, localized versions) throughout the entire Roman-influenced world, except for places the Arabs later overwhelm with their influence.
A much weirder and in effect ASB scenario would be if the Hebrew or Aramaic scripts were adopted in a fit of Christian piety.