Who was the recordkeeper in 3 Nephi? Probably Nephi, the high priest, I would say. When his father left, he handed over to Nephi all the sacred records. He had been trained as a recordkeeper. On a later visit, Jesus asked Nephi to bring the record that had been made of Samuel the Lamanite’s prophecy. Can you imagine the Savior proof-reading your work and saying, “You know, you are pretty good at this, but you missed something”? Of course, Nephi fixed it right away. He was the recordkeeper. While we do not know whether the words of Jesus were recorded in real time, they were kept diligently and promptly.
Even if his words were not written as they were spoken, the disciples no doubt remembered it soon enough that night. The ancient people had very good memories because their minds were not as cluttered as our minds are today. They were in the habit of memorizing, and it was not uncommon in ancient pedagogy for a teacher to simply talk and then the students would recite what the teacher had said. Today, we have recording devices, and because of that, we are lazy. Without that technology, people actually listened and remembered, but they would write it down as soon as possible so that it would be available accurately. That is probably why we have exactly the words that Jesus spoke here in 3 Nephi.