Christianisation of Barbaricum proceeds as OTL, with more and more Germanic peoples being taught to use a Greek alphabet to read their bibles in the native language.
The main problem is that translation of the Bibles were at some point more a prestigious device than an actual alphabetisation of the germanic peoples.
That alone wouldn't have removed a possibility of Goths using a Greek alphabet more long that they used Gothic script, but in the same way germanic peoples converted to Orthodoxy (or converted back, as for Burgundians), the fusion between German and western Roman elements that used Latin script including for religious devices since too long to be rooted out would eventually see Latin script being favoured.
As long scholars of romano-german kingdoms are Romans or at least latinists, Germanic culture and script wouldn't have a great chance to impose.
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The most obvious PoD would be a Roman-Screw. Should Rome never manages to really go outside Italy, maybe with a Punic victory, Greek script would have a quite better chance.
Rome, Carthage were quite hellenized already, and would probably continue to do so more intensivlty, maybe not to the point of adopting a Greek script for their languages but not posing a cultural threat to "alphabetizing" of non-scripted languages.
You may see more occurence as OTL Gaul adoption of Greek script to write Gaulish, depending of Greek or Hellenized trade presence in Europe.
I would tend to think, however, that without a political hegemony akin to Rome's; Greek wouldn't have a same impact. The fact Roman Empire, a sole political and economical continuum, used Latin as an administrative language really helped its diffusion.
TTL may see the rize of native script as Futhark, but more influenced by Greek Script rather than Latin. Eventually, not a dominance of Greek Script per se, but a more important influence.