The Quran asserts that Jesus' apostles called themselves Muslims:
When Jesus found Unbelief on their part He said: "Who will be My helpers to (the work of) Allah?" Said the disciples: "We are Allah's helpers: We believe in Allah, and do thou bear witness that we are Muslims. "Our Lord! we believe in what Thou hast revealed, and we follow the Messenger; then write us down among those who bear witness."(Sura 3/52-53)
Although the Quran has no description of how the followers of Jesus went from "bearing witness that we are Muslims," Sunni Muslims accept as valid an exegesis of the Quran by Ibn Kathir (1301-1373 AD), which would be considered a Hadith (a saying of Muhammad as conveyed verbally through someone close to him):
Ibn Abbas said, "Just before Allah raised Jesus to the Heavens, Jesus went to his disciples, who were twelve inside the house. When he arrived, his hair was dripping with water (as if he had just had a bath) and he said, 'There are those among you who will disbelieve in me twelve times after you had believed in me.' He then asked, 'Who among you will volunteer for his appearance to be transformed into mine, and be killed in my place. Whoever volunteers for that, he will be with me (in Paradise).' One of the youngest ones among them volunteered, but Jesus asked him to sit down. Jesus asked again for a volunteer, and the same young man volunteered and Jesus asked him to sit down again. Then the young man volunteered a third time and Jesus said, 'You will be that man,' and the resemblance of Jesus was cast over that man while Jesus ascended to Heaven from a hole in the roof of the house. When the Jews came looking for Jesus, they found that young man and crucified him. Some of Jesus' followers disbelieved in him twelve times after they had believed in him. They then divided into three groups. One group, the Jacobites, said, 'Allah remained with us as long as He willed and then ascended to Heaven.' Another group, the Nestorians, said, 'The son of Allah was with us as long as he willed and Allah took him to Heaven.' Another group, the Muslims, said, 'The servant and Messenger of Allah remained with us as long as Allah willed, and Allah then took him to Him.' The two disbelieving groups cooperated against the Muslim group and they killed them. Ever since that happened, Islam was then veiled until Allah sent Muhammad."
Ibn Abbas, by Wikipedia, was a paternal cousin of Muhammad. This "valid" Hadith is notable for its inclusion in the first century of Christian sects that originated centuries after Jesus lived.
The Hadith refers to the traitor followers of Jesus who proclaimed him as the son of God as "Nestorians." This is a huge chronological mistake. Nestorianism was a doctrine advanced by Nestorius, Patriarch of Constantinople between 428-431 AD. There's no way that a group of Nestorians would have been present among Jesus' disciples 400 years earlier. It's interesting that Muslims associate this sect of Christianity with apostates from Jesus, because it's very similar to the doctrine of the major Christian church in the Sassanid Empire, which Muslims conquered: the Church of the East. The Church has surviving members today in Iraq, Iran, and Syria. In the western churches, Orthodox, Catholic, and Protestant, Nestorianism is considered a heretical doctrine that claimed the divine and human natures of Jesus Christ were totally distinct: "Jesus Christ, who is not identical with the Son but personally united with the Son, who lives in him, is one hypostasis and one nature: human."
The Hadith also refers to apostate followers of Jesus who proclaimed him to God himself as "Jacobites." This is another huge chronological mistake. Jacobites were named for Jacob Baradaeus, Bishop of Edessa (Turkey), 543-578 AD. Modern followers of this doctrine are in the Syriac Orthodox Church, who believe their foundation in Antioch by Peter. "Jacobism" was historically condemned by western churches as a form of miaphysitism, which believes that Jesus' divinity and humanity are combined in a single nature. Jacobites existed 500 years after Jesus lived. And no surprise, the Muslim armies conquered the territories in which these Christians lived. Notably, the Jacobites felt that Jesus had divinity, and that's reflected in the Hadith's description of their "apostasy."
Looking at how the Hadith cites multiple groups of Christians as "disbelieving groups" that killed the true-believing Muslim followers of Jesus, it seems quite notable it cites Christians living in different parts of the Middle East during Muslim conquest. The Hadith appears to bebased on back-projecting Muslim understanding of Middle Eastern Christians (which seem to reflect the actual beliefs these people had) into the 1st century.
That the Hadith uses descriptions of conquered people to condemn Christians is probably an effective way to win converts. "Look, we beat them!" is a great way to convince the ignorant that your religion's better than that of the land you just conquered. As I'll address later, the Quran itself plays on Christian misfortune to suggest that the worldly woes of Christians are due to their apostasy from Islam.
Hadith aside, the Quran itself says that the followers of Jesus called themselves Muslims. So what is the evidence for the change to "Christian?"
Using exclusively extra-biblical sources, this seems an easy one to prove. Pliny the Younger was governor of the Roman provinces of Pontus and Bithynia from 111-113 AD. He persecuted Christians, and wrote of them to the Roman Emperor Trajan:
"they were accustomed to meet on a fixed day before dawn and sing responsively a hymn to Christ as to a god, and to bind themselves by oath, not to some crime, but not to commit fraud, theft, or adultery, not falsify their trust, nor to refuse to return a trust when called upon to do so. When this was over, it was their custom to depart and to assemble again to partake of food--but ordinary and innocent food. "
Pliny describes Christians as committed to a moral life and to eat together, probably in the *agape feasts (the early Eucharist). Granted, this was ~70 years after Jesus lived, but this sound like modern Christianity. Pliny was a Roman Pagan, not a Christian.
In another extra-Biblical source, we have the Christian first letter of Clement to the Corinthians (80-140 AD):
"Jesus Christ our Lord gave His blood for us by the will of God; His flesh for our flesh, and His soul for our souls."
This statement sounds remarkably like what modern Christians say.
The Himyarites were a group of Arabs living in Yemen whose rulers had converted to Judaism. They'd conquered their neighboring state in 25 BC, "Saba" ("Sheba" in the Old Testament). Many Christians lived in the region. In 525 AD, the Arian Christian ruler of Ethiopia invaded an put a Arian Christian ruler on the throne. Also relevant to the story, there was an ancient dam at Marib that allowed extensive agriculture in the arid region built by the previous rules of Saba. In 325 AD, the dam was expanded to allow the irrigation of 25,000 acres. However, river sediment kept collecting on the bottom of the reservoir it held, and though the Himyarites tried to raise the height of the dam, water kept flowing over the top. Apparently some of the ancient knowledge on managing the dam was lost as time went on. In 449, 450, 542 and 548 there were major flooding incidents, with the last recorded repairs took place in 557. In 570 or 575, the dam was overtopped and catastrophically failed, causing massive flooding throughout the region and causing the migration of tens of thousands of people throughout Arabia. Many of the Christians moved up in to Jordan, and other refugees carried word of the flood throughout the region.
That the dam was run by a kindgom conquered by Arian Christians was not overlooked in the Quran:*
"But they turned away [from Allah], and We sent against them the Flood [released] from the dams, and We converted their two garden [rows] into "gardens" producing bitter fruit, and tamarisks, and some few [stunted] Lote-trees. That was the Requital We gave them because they ungratefully rejected Faith: And never do We give [such] requital except to such as are ungrateful rejecters."*(Sura 34/15-17)
Here is a catastrophe shocking all Arabia a century before Muhammad in the Quran's depiction of those who reject Allah. Again, this looks like back-casting. A flood happened not because of too such sediment, which the warring rulers didn't have the resources to manage, but because they were Christians.
Now, assume for a second that you're an atheist historian. Still, you're writing a book on the history of Christianity. If you were going to cite historical documentation for the practices of early Christians, you would sound much more credible if you cited historical sources that were closer in time and geography to what those early Christians practiced. An atheist historian wouldn't cite the Quran as historical evidence of what early Christianity. Someone would believe the Quran over all the other evidence only because they want to believe the Quran is right. To me, this is a case where my "faith and reason" (fnr) are in agreement. To believe that the disciples of Jesus were Muslims, I'd have to reject so much evidence to the contrary as to discredit myself to any historian.