Good question. They would be closer to OTL than you think.
For one, unless you posit a huge disintegration of the Eastern empire, you still have Eastern Orthodoxy. ALL of the ecumenical councils in the early church were called by the EASTERN emperor and not considered binding unless the Eastern emperor ratified them (sorry Roman primacy types!)
Plus, even with the disintegration of Western Roman power, you still have Rome itself being an important city, with the Roman Bishop being a powerful figure somewhere there is a power vacuum. So, voila, you have ATL Roman Catholicism as well. You had a Bishop of Rome trying to flex jurisdictional might since the late 2nd century (Easter controversy), so your POD would actually have to be before the late 100s in order to make a different church provided that you include in your POD that Christianity becomes the dominant religion in former Roman territory.
But, this begs the question--how on Earth does Christianity become a majority former-Roman religion without a state entity imposing it? Islam is big because a state entity imposed it. The same is true of Christianity in Christian lands. It was literally against the law not to be a Catholic/Orthodox Christian. How do you get large scale Christianity without Roman or insert state entity here power?
Being that I don't want to ignore the whole point of this thread, I will say that if you want to avoid the ATL turning out almost identical to OTL, you would need a POD either very early (literally Roman disintegration before Christianity and a slow, but sure climb to dominance like Buddhism in south east asia) or Christianity imposed by Germanic tribes.
In the former, the Christianity apart from state actors would probably look a lot like modern Protestantism. It would have some level of doctrinal cohesion, but not like a state church.
In the latter, Christianity would be almost impossible to recognize. It would be essentially a Germanic Odin-Thor religion given new clothes. But, the Germans were never going to be able to take over Greece and Egypt. So, you would essentially end up with Eastern Orthodoxy and Arianism in the West, probably centered in Rome with a Pope.
If you want to avoid the existence of Eastern Orthodoxy with a post Jesus Christ Roman break-up, you would need something absolutely cataclysmic which would fragment Roman control in the are. But, who takes over? Were there enough Greeks to reassert themselves? WOuld it be like the post-Alexander the Great break up of the Persian Empire? In that case, you would have a different Christianity in each sphere suited to the whims of the emperor of the fragment of Greece/Egypt/Persia/Arabia/etcetera. If this occurs, it's anyone's guess how it develops.
I will say this--the Roman/Eastern/Oriental system of a priesthood, as long as there are state actors, is probably going to become solidified. This is not because states invented the priesthood (it was pretty much there by the late 100s early 200s.) However, the priesthood fit cultural norms of the time (every religion, including Judaism, had them) and it is only natural, when there are state actors involved, that the chief priests in the big cities are going to exert disordinate power and own more properties (which means, more power.) In this sort of enviornment, an monarchical episcopacy is pragmatic and a forgone conclusion.
The plurality of Bishops, which even Catholic historians admit was the original practice of the Apostolic Church, simply is too counter-cultural to last and IOTL it had pretty much disappeared by Ignatius' time.