There was no sense among the early Christians that anything about Jesus, his discourses and their meaning needed to be set down in writing until the apostles began to die off. The first apostle to die was James the Greater, martyred in the year 44 of our era (Acts 12:1-2); the last was John, in Ephesus in 98.
Sometime in between the chief eyewitnesses of Jesus earthly ministry passed on the basis of the writings we know as the New Testament.
Almost certainly, the texts began with preaching that was only recorded or recounted to scribes as it became clear that the eyewitnesses would pass away before all of Jesus' prophecies became true. Except for Paul, the apostles were not intellectuals and writing probably would not come easy for them; in any case, most of the first Christians were illiterate or only semi-literate.
Paul was the earliest author of writings in the collection. As many as 14 letters were attributed to him, loosely or expressly; today seven pass muster as unquestionably the actual work of Paul, the remainder penned by close associates and deemed representative of Paul's teaching.
The four accounts we know as gospels were written 20 to 90 years after the crucifixion. Mark was at least drafted in Rome while Peter, from whom it draws partial inspiration, was still alive. Luke, drawing from Paul, and Matthew, associated with the apostle of that name who resided with the Antioch community, may not have reached final form until as late as 110 or 120.
Acts, the first history of Christianity, is also the work of Luke.
Then there are a series of letters by followers of other apostles. The NT concludes with the very strange, mythical and visionary book of Revelation or Apocalypse, attributed to the apostle John but almost certainly not his work.
As I warned with Old Testament or Hebrew Bible, one should not conclude that the uncertain authorship makes these writings fake. Authorship in the ancient world was a far cry from what we, in the copyright era, deem it to be.
Authorship could mean that someone had dictated something to a scribe and that some editor had taken it in hand and given it literary shape. It could also mean that someone in a school of thought had attributed a summary of its ideas to one of its principal proponents. Notably, not one word attributed to Socrates was written by Socrates.
Importantly also, just as the OT did not drop out of the sky translated into 17th century English and leather bound by Thomas Nelson, the NT was not instantly approved for reading from lecterns by the Church. Indeed, although there are nearly identical listings of NT books, notably that of third century Christian writer Origen, in various documents, none of these lists was an official statement.
As we shall see in the next few centuries of intense debates, such declarations have usually been made to exclude what plainly was seen not to mesh with the received doctrine or teaching about the faith. No one thought to challenge the canon—or list of books in authoritative scripture—until an Augustinian monk who suffered from bad digestion, one Martin Luther, came along.
Thus, although by all accounts the Bible effectively closed around the year 120, there was no official statement naming and numbering the books of the New Testament until the 16th century. These include the Council of Trent's decree in 1546 (Catholicism), the Gallic Confession of Faith of 1559 (Calvinism), the 1563 Thirty-Nine Articles (Church of England) and the 1672 Synod of Jerusalem (Greek Orthodox).
However, through the centuries from the point at which the ink first dried on the scroll containing the Apocalypse, there came to be established translations that were widely used. Notably, the 4th century translation to Latin by St. Jerome, known as the Vulgate, was by the early Middle Ages found in every church in Europe.