How can we have the “exact words” of Jesus, e.g. during His interrogation by Pilate or when He prayed alone in Gethsemane, when none of his disciples/followers were present to hear His very words?
First of all, we do not have the “exact words” of Jesus in our Bibles. The reason for this is that the originals of the gospels were in Greek, as far as we know. I assume that the conversation between Pilate and Jesus as well as Jesus’ prayers in Gethsemane were in Aramaic, so, by definition, we do not have the exact conversations. At best, they are translations of the words spoken.
Having said that, it is entirely possible that Jesus shared the contents of his conversations with his disciples after he rose from the dead. Besides, when he breathed on the apostles in John 20:22, he gave them miraculous/inspired knowledge as well. Let me be honest, I am not sure exactly how the apostles came to know the content of the conversations of Jesus with Pilate and in the garden. We cannot know because the Bible does not tell us how the gospel writers got this information. It is entirely possible that there were others in the palace with Pilate who heard what was said. You say that “none of his followers were present” at these events. This is an assumption, but it may not be a correct assumption. It would not be surprising if there were others who heard him in the Garden of Gethsemane as well. As I already said, Jesus may have shared the details with them. Being God, he knew what his Father intended to have end up in the scriptures, which could easily explain what he told them during the many times they met between his resurrection and his ascension. We simply do not know.
What we do know is that the Bible is inspired by God. The evidence for this is extremely strong. I believe that some of the conversations of Jesus in the gospels are good-faith reconstructions by the apostles of actual conversations Jesus had. They do not have to be word-for-word in order for them to be inspired. The question is not whether the conversations are word-for-word, but if they are faithful to what Jesus said and are inspired by the Holy Spirit. I believe they are, based on the evidence for biblical inspiration, but of course everyone must decide for themselves what they believe.