The assumption behind this question is that Christianity was originally based in Rome instead of Jerusalem. I would argue that that's not true, certainly not when Christianity was a new and distinct religious identity. Jerusalem remained a major location for Christianity up until its total destruction between the AD 70 Roman-Jewish War and the AD 135 Bar Kokhba revolt. For the next couple of centuries, the two most prominent centers of Christian learning and leadership were Alexandria (Egypt) and Antioch (Asia Minor/modern Turkey). With the ascension of Constantine as emperor, the legalization of Christianity, and the establishment of Constantinople as an imperial capitol, that city likewise became prominent. While there were Christians in Rome, that city doesn't start to play a major role in ecclesiastical politics until the middle of the 5th century when Pope Leo I got involved with the Council of Chalcedon.

So unless you consider a gap of 400+ years to be within "the brand new church of Christianity," I'd characterize Rome's prominence as a later development. Through the 6th century, there was more of a collegial view to the five major cities and their church leaders: Constantinople, Alexandria, Antioch, Jerusalem, and Rome.

Politically, the real division comes with the emerging division between the Eastern and Western parts of the Roman Empire. The division had always existed to some degree in cultural and linguistic terms, the East dominated by Greek and the West more by Latin. This took on a political reality starting under Diocletian around the turn of the 4th century but grew until by the end of the 5th century we talk about "the fall of Rome," meaning the collapse of the Western Roman Empire. The Eastern Empire kept ticking along and while today we often refer to them as the Byzantine Empire, they considered themselves Roman, albeit based in Constantinople over a largely Greek Mediterranean area.

This importance of this for Rome lies in those patriarchal cities: four of the five are in the Eastern Empire, while only Rome is in the Western area. As political links and communication faded, this left Rome as the only historic option in that territory, and as the socio-political-linguistic divide deepened, the ecclesiastical divide did as well. But throughout all of Christian history, it's never been accurate to say that the entire Christian religion was based in Rome; the Western European church was based in Rome, but the Eastern Orthodox church has always remained in that area -- not to mention a number of other churches just as old or older than Rome, like the Coptic Orthodox in Egypt, the Ethiopian church, the Armenian church, and others.