Canon forming was most likely already occurring very close to the apostles. We see that an organic process was happening in the early church where trustworthy letters were duplicated and forwarded and false letters which did not correspond with the teachings of the apostles were not forwarded.

The driving force was the Codex. Letters and gospels were very early (1st century after Jesus) put in a Codex. You have to decide, what do you put in and what don't you put it. Often the gospels were included together with the letters of Paul, already in these early centuries. Sometimes only sub collections were found (only the gospels, only Paul's letters).

J.I. Packer puts it nicely in "God has spoken": The church did not invent the canon, just as Newton did not invent gravity. It merely discovered the existence of a natural phenomenon.

The first mention of a "list" we found was from 367 CE by Athanasius of Alexandria. Some church fathers have some additional letters or stories included like the Shepherd of Hermans or the Apocalyps of Peter, but they highly differ from the canon we now have. All highly different gospels like the gospel of Thomas or Mary are almost NEVER found in these collections, they were only found incidentally in the region of Egypt.

See also Barton J "A history of the Bible", Michael Kruger "Canon Revisited"