There's a theory spreading in Twitter that Hwan is already in time loops throughout the drama, but I disagree with it. I believe this drama shows the original timeline of the first time they met before Hwan was given the ability to time travel. It provides their introduction, backstories and build up before the time loops starts. That's why Begins ≠ Youth acts as a prequel to the Save Me webtoon.
In the intro scenes of the episodes, it's centered around the fire incident of Cein's mother and how the other boys are involved to it. The time loops are occurring there with a suspicious man. It may look like a younger guy, but I think it's either Hwan's father or the dead man in the beach (Byeon Wonjoong) who was one of his friends from the past. Actually, Hwan's father did time loops in the past to save his friends (the students in the scholarship program) but he failed so he lost his memories. While Byeon Wonjoong revealed to Jooan's father how he died several times which implies that he also went through time loops recently. The purpose of the adults' time loops is to save their own friends, which includes Cein's mother. That's why he planned to prevent the fire by using the boys and giving them orders. But at the end, Cein's mother still died in another way. Everyone involved in that fire incident forgot their memories since it failed (Jooan also met Cein before but they don't seem to have a clue, like everyone else). It's impossible that the suspicious man is Hwan since the fire incident happened a year before he arrived from the US. During the finale, it exactly shows how Hwan went back in the past on the day he first arrived in Korea. (Then the title card appears, Begins ≠ Youth changing to Begins ∞ Youth with the infinite symbol which means his first time loop is now only beginning.)
While for the diary (The Note), it exactly shows that it's owned by Byeon Wonjoong. So this concludes that Byeon Wonjoong is possibly the suspicious man/time traveler from the fire incident and he went through multiple time loops, where there might also be a timeline where he followed the boys in their future that's why he knew about their fates. He also wrote about Hwan as the "Boy of Fate". I think he's trying to help them since the adults failed on their time loops.
In the original book, Hwan found the diary of his father. But in the drama, Jeha found the diary of Byeon Wonjoong. That doesn't mean Jeha has the same time travel ability as Hwan. But he is aware of the events happening and timelines changing. While Jooan has the ability to see the future on his dreams, but he is unaware of it at first. (Jooan's father was also friends with Hwan's father and Byeon Wonjoong in the past.)
There's been many changes including mixing up the events in the drama. But in the original ending of the book, Hwan succeeds to save them all and ended his time loops after finding his "Map of the Soul". But something is still wrong with Jeha because he thinks Hwan did something bad to him and he feels indifferent from the group. Due to his confusions and suspicions with the events and timelines, he is stuck in a wormhole. Apparently, it seems like Jeha also tried to make a contract with the cat (the "being" who gave Hwan the ability to time travel, but it was only a voice and nothing was showed in the drama). Overall, the BU storyline is not really complete and still leaves with an open ending.
In the drama, it ends with Hwan going back in the past. He will be stuck in multiple time loops to save the boys, prevent their tragedies and change their fates. There are still many things to happen in the storyline and I don't know how this drama will add it, but I still keep on hoping that there's Season 2 so they can show and explore more about Hwan's time loops along with the new timelines.
Disclaimer: The BU storyline, especially the time loop parts, is very long and confusing. It's normal that everyone has their own theories and interpretations. I could also be wrong about mine. At the end, there's been many changes in the drama and we can't expect that they will follow the original sources from the book/webtoon.
