This answer is a bit different for the Roman Empire and for the early Christians themselves.

The Romans seem to have decided that Christans were not just a sect of Judaism pretty early on. A letter of Pliny the Younger to the emperor Trajan in AD 112 when the former was the governor of Pontus and Bithynia is frequently cited. The first part reads:

"It is my custom, Sir, to refer to you in all cases where I do not feel sure, for who can better direct my doubts or inform my ignorance? I have never been present at any legal examination of the Christians, and I do not know, therefore, what are the usual penalties passed upon them, or the limits of those penalties, or how searching an inquiry should be made. I have hesitated a great deal in considering whether any distinctions should be drawn according to the ages of the accused; whether the weak should be punished as severely as the more robust; whether if they renounce their faith they should be pardoned, or whether the man who has once been a Christian should gain nothing by recanting; whether the name itself, even though otherwise innocent of crime, should be punished, or only the crimes that gather round it."

The letter goes on for several more paragraphs, and we have Trajan's response as well, telling Pliny not to seek out Christians for punishment, but to punish any who refused to recant their faith.

Not once in either letter does the word "Jew" appear. This is significant. The Jews had an unusual place carved out in Roman law. The Romans required the populace to honor the Emperor as divine and to make sacrifices to his cult. For most of the polytheistic inhabitants of Roman territory, this was no big deal - what was one more god amongst many? However, the Jews were fiercely monotheistic and were commanded to have no other god but God. Showing a fair degree of pragmatism, the Romans decided to exempt the Jews from worshipping the emperor on the basis that theirs was an ancient faith that predated Rome itself. This exemption kept the Jewish lands peaceful, for the most part (see below).

Pliny's letter clearly shows that the Roman view was that Christians were worshipping Christ, who had been a living man not far removed from then-current times. There was no ancient basis to their faith, and they were viewed as merely a new cult that also refused to worship the emperor. Pliny's letter is the earliest surviving written account of the attitude of the Romans to the new religion.

Both Tacitus and Suetonius wrote about earlier interactions with Christians, however both accounts were written after Pliny's letter.

Tacitus wrote about the response to the great Roman fire of AD 64:

Nero fastened the guilt and inflicted the most exquisite tortures on a class hated for their abominations, called Chrestians by the populace. Christus, from whom the name had its origin, suffered the extreme penalty during the reign of Tiberius at the hands of one of our procurators, Pontius Pilatus, and a most mischievous superstition, thus checked for the moment, again broke out not only in Judæa, the first source of the evil, but even in Rome, where all things hideous and shameful from every part of the world find their centre and become popular. (Annals 15.44)

Again, no mention is made of "Jews" and it was clear that these people followed "Christus". So sometime between the death of Christ and the accounts of Pliny and Tacitus, the Romans came to view Christianity as something other than a Jewish sect. The Christians might have helped fuel this view, as their religious practices were mostly secretive and held at night to avoid the authorities.

For the Christians themselves, the question is maybe harder. The earliest Christians remained Jews in their own minds, continued to pray in the Temple, and maintained their Jewish identity. The first challenge to this belief came when converts came from outside Jewish lands, particularly in the city of Antioch. For these new converts, the message of Christ resonated, but there was no Jewish identity or culture for the largely Greek-speaking people hearing it. Early Christian leaders debated whether these new converts needed to be Jews first (be circumcised and accept Jewish dietary law, amongst other things) before being considered members of the Christian community (the answer wound up being "no"). As time went on, a greater and greater number of Christians were Gentile converts. These men and women were never seen as Jews at all. Simultaneously the Jewish leaders began to officially expel Christians from Jewish life.

The situation reached a final head during the Bar Kochba revolt (AD 132-135), during which rebels expelled the Romans from Judaea and declared a short lived new Jewish kingdom under the leadership of Simon bar Kochba. During the revolt, Christian writings were outlawed, and perhaps more importantly, Bar Kochba was proclaimed the Messiah by the most influential rabbi of the time, Akiva. That was the final blow for remaining Jewish Christians, who already had their own Messiah and din't need another.

So it seems that in the mid to late first century AD most Christians would have considered themselves Jews, but this changed significantly over the next 50 years or so. At the same time, the Romans were more inclined to always view them as their own, new religion. This process gained increasing steam as the number of Gentile adherents soared.

See:

Lost Christianities (Bart Ehrman)

The Christians as the Romans Saw Them (Robert Lois Wilken)

The Letters of Pliny the Younger (Betty Radice)

The break between Judaism and Christianity was a complex process and there is no single date we can pinpoint, scholars of early Christianity and post Temple Judaism call this the "parting of ways". Already within the first decade or so, gentiles (non-Jews) were beginning to join the religion, but this actually wasn't all that new to Judaism as there had already been gentile (God-Fearers) participating in Hellenistic (Greek) Jewish communities before the advent of Christianity. With the growing number of gentiles joining the early Jesus movement, internal debates began to erupt over whether or not these converts had to undergo a formal conversion into Judaism (via circumcision) and this is where we see the conflict between the so called Judaizers and Pauline Christians come into play. I should mention that Paul himself was a Jew and a Pharisee (and never actually stopped claiming to be one for the record) and his relationship to Judaism as a whole is actually quite complex and cannot be covered in a single post.

Anyway, by the end of the 1st century (A.D. 96) the Romans, under Emperor Nerva, exempted Christians from having to pay the Fiscus Judaicus, a tax levied by the Romans on the Jewish people across the empire as repriations for the First Roman-Jewish war (A.D. 66 - A.D. 73), and it also served as a stand in for having to participate in the Roman imperial cult which the Jews could not due to the monotheistic nature of their religion. This created a lot of problems for the early Christian community once they were exempt because it meant that the Romans considered Christianity distinct enough from the rest of Judaism to consider it a cult of its own, and it also meant that they were no longer exempt from having to participate in the Imperial cult. This is, as u/Frescanation already posted, where the correspondences about Christians between Trajan and Pliny the Younger come in during the early 2nd century.

But this distinction was not limited to only the Romans, but actually among many of the Christians themselves. Ignatius of Antioch, a 2nd century Christian bishop, is the first to make the distinction between those who "practice Christianity" and those who "practice Judaism". There had been a concerted effort among some of the early Christian clergy to make a theological/sectarian distinction between Jews and Christians. Not all Christians saw it that way though, and even as late as the 4th century there were still Christians participating in Jewish communities to the point that St. John Chrysostom wrote eight homilies polemicizing against the Jews and warning Christians away from going to their synagogues. It should be noted, however, that the last major Jewish-Christian community that could be traced to the original Jesus movement existed in Jerusalem in direct continuity to Jesus's sibling, James the Just. But this community was destroyed during the Bar Kokhba revolt in A.D. 132 as a result of them not accepting who was then the rabbinically approved Messiah Simon ben Koseveh.

I recommend you read: The Parting of Ways by the late James Dunn on this subject. :)