This is more of an AHC. The effects of the Roman bishopric not being that important are unpredictable.
The problem here is that at the time of the Council of Nicaea, the eastern half of the Roman empire was much more Christianized than the western half. Nearly all of the participating clergy came from the eastern half. The city of Rome itself was just important enough, and the western half just important enough, to get one of the five patriarchs but not Christianized enough for two.
So to keep the Patriarch of the West from being the Bishop of Rome, you really need to screw with the time line in ways that would produce massive butterflies:
1. You could have Christianity make so little progress in the west that there is no western patriarch, but now you have changed the history of early Christianity and you probably have gotten rid of either Constantine's conversion or his conquest of the Roman Empire. This also gets rid of the Council of Nicaea itself.
2. You can make the Western half much more Christian, but again this changes the early Christianity, and there is still the problem that the second western patriarch is much more likely to be in the areas taken over by Islam (North Africa or Spain) than in Gaul.
3. Most feasible is to have one western patriarch, but the location is a different city, most likely Milan. In this case history unfolds as IOTL, just with the Pope based in Milan and not Rome.
Now #3 is a realistic POD and while you still get one Pope in the Western Church, the history of the Papacy starts to diverge from the OTL history of the Papacy for several reasons. First, Rome after much of its sanitation infrastructure was destroyed in the sixth century, was malarial and you had lots of Popes dying at inopportune times that wouldn't happen if the Papacy was in Milan, or trying to leave Rome which again wouldn't happen. Second, the Byzantines were actually pretty successful in controlling the Popes in the sixth and seventh centuries and for geographical reasons it would have been harder to project power to Milan. Third, a completely different local aristocracy. So butterflies would accumulate but you still get one line of Popes in the Western Church, just different individual Popes.