This is a deep and fascinating question, so I'll try to pare it down to the essentials.

The early Christian church was a fairly decentralized organization. Local Bishops were largely autonomous, with major issues being decided at major church councils. The five most prestigious centers (and therefore Bishops) of Christianity were Antioch, Alexandria, Jerusalem, Rome, and Constantinople - these formed the Pentarchy; in theory, a sort of republican system to decide major church matters.

In practice, Rome considered itself preeminent, which they thought meant they should be in charge, while Constantinople gradually overtook the other three in prestige, and argued that Rome was merely 'first among equals.'

Eventually, Antioch, Alexandria, and Jerusalem were lost to the Muslims, and it was just Rome and Constantinople. Then the King of Bulgaria decided to convert to Christianity. The Christianization of the Bulgarians is a fascinating subject in itself, but the long and the short of it is Rome and Constantinople fought over who the Bulgarians would be under. The King played them off each other, and eventually had Constantinople recognize the head of the Bulgarian church as first an archbishopric, and then later a Patriarch.

This latter move was incredibly controversial. The only patriarchs up until now were the Pentarchy, and by calling the head of the Bulgarian church a Patriarch, Constantinople had basically elevated them to the same status as Constantinople and Rome. The Pope flipped, arguing that Constantinople didn't have the authority to do that.

Since the Patriarch of Bulgaria was at the same level, rather than below, the Patriarch of the East (Constantinople), they were considered 'autocephalous' - the head of their own branch of Christianity, which became known as Bulgarian Orthodox. This pattern followed for other Orthodox groups, naming themselves after the people their missions were to.

Now, to answer your question a little more directly (once all that background is out of the way) - nationalism didn't emerge until the 19th century, but that doesn't mean that nations didn't. There were Germans, and Slavs, and Bulgarians before there was a Germany or a Bulgaria. Nationalism was more the idea that these larger identities should be a single state - a nation state. One of the main factors in the Christianization of the Bulgarians is that the King of the Bulgars realized he had two very different groups - Slavs, and Turkic Bulgars, who had different languages and religions. He used Christianity to help forge a single identity, and as the Bulgars Slavicized, their (now single) religion was a force of cohesion.

A big part of this is the lack of a single figure, equivalent of say the Pope in Catholicism, in charge of the orthodox church. Instead Orthodox churches were ruled by Patriachs who managed areas of orthodox lands. For example there was a patriach in Constantinople, who admittedly may have been the first among equals in the system at one point, known as the Ecumenical Patriach.

But perhaps the main reason this occurred is the split of the Roman Empire and the collapse of the Western Empire. While at first there was no political issue with having the Pope remain in Rome in the Western half of the empire, the ecumenical Patriarch developed for largely a similar reason as the split in the empires, the empire was too large for one emperor and similarly it was too large for one Pope. As such it made sense for other important figures to reside in the capital of the Eastern half too. The real split between the churches occurred with the fall of the Western Empire. The Pope in Rome continued to insist that he was the head of the Catholic faith. For the Eastern Romans (Byzantines from here on) this was largely unacceptable, religion and state were intricately linked and the emperor couldn't really afford to be beholden to an independent Papacy not to mention the fact that the same administrative difficulties remained as before, having all religious issues go to and from Rome would seem a waste of time when there is a perfectly good religious leader in the same city as you.

After this point, orthodox religion kept spreading East with the rise of Rus' and remained in lands in the Middle East for some time after Muslim conquests. So these two factors, the spread of the faith and the decline of Byzantium meant that it again became practical to have local religious leaders take on increasingly important roles and become more and more independent.

These churches wouldn't have been called Russian Orthodox, Greek Orthodox, etc. until the rise of states bearing those names but there was a Patriach in Kiev, which later transferred to Moscow and this would go on to become the Russian Orthodox Faith. Greek Orthodox meanwhile is a more general term for former Byzantine churches as the Byzantines were increasingly called Greek by Catholics, this is why it also includes churches such as the Jerusalem Patriachy.

Source is just my knowledge from relevant history course at Uni so feel free to correct me but I thought I'd try and help.