This will necessarily be a little on the high concept side of things, because of the nature of many of the Germanic tribes’ conversion to Christianity: those that embraced the Arian Creed were generally converted by the Romans when Arians were governing the Empire. So, for the tribes to adapt the Nicene Creed, we’d have to change a lot of small details that could add up quickly.
So, lets not worry about that... which is the AH equivalent of saying ‘don’t think about elephants.’ Anyway, if we assume the broad strokes are the same, how might this play out as Goths and Vandals and others settle across the Empire? In particular, what might this mean for the longevity of the Empire?
On the one hand, you have the school of thought that the new blood could reinvigorate the Empire and would now be far more likely to be accepted among the population. They’d be looking to the same Church leaders for guidance, and within a few generations could be knitted into the fabric seamlessly. There would be little notion of seperate identities for Romans and Goths.
On the other hand, the tribes could still maintain their distinctive identity, but now, without a religious divide between them and the population living in their new lands, there’s even less reason to prefer Imperial rule to Germanic rule. Sure, they might look to the Pope for spiritual guidance, but all it takes is one Pope that prefers the protection of a pious Gothic king to an overbearing Roman Emperor, and that common ground can help tear things apart.
Or perhaps something in the middle? Something more loosely knit, like the Holy Roman Empire of later centuries? Just throwing a bone to anyone that doesn’t like the first two options.