His relationship with his family might become more protective. They would worry about him more and might be overbearing at times. For example, his brothers might not let him go on dangerous missions alone anymore.
In a romantic relationship, if he was in one, his partner would have to be more understanding and patient. There would be more worry and care involved. They might have to adapt their lifestyle together, like avoiding places where there are a lot of germs to protect Tim's weakened immune system.
With his friends, they might see him as more vulnerable. Some friends could be extra helpful, while others might distance themselves because they don't want to get involved in his new - found health issues. It could cause a rift in some of his friendships.
One idea could be that Tim has to constantly be careful about his health now. Maybe he has to take special medications and his crime - fighting activities are restricted. This creates a lot of internal conflict for him as he doesn't want to be sidelined.
With the other Robins, like Dick Grayson and Jason Todd, his relationships would vary. If he was from a more violent universe, he might not get along with Dick's more positive and acrobatic approach. And he could have a more competitive or even hostile relationship with Jason if they were both vying for the same territory or resources in Gotham.
It would likely make his relationship with Iruka even more important. Iruka was one of the first to show Naruto kindness, and with his parents' absence, he would cling to Iruka as a father figure even more. Naruto might also be more eager to prove himself to his peers. He would want to show that he doesn't need his parents and can be just as good or better than those with a normal family. This could lead to some over - competitiveness at times.
His development would likely be centered around his identity struggle. On one hand, he has his human side, but on the other, he has the angel - influenced side. This internal conflict could make his character growth more complex. He may constantly question his actions and motives, wondering if they are influenced by his human nature or the angelic upbringing. This could lead to a more nuanced and evolving character over the course of the fanfiction.
His superhero identity might give him access to exclusive Mets events. For example, he could use his connections in the hero world to get backstage passes to meet the players.
Naruto's relationship with his teachers could be different too. Kakashi, for example, would need to adjust his training methods for Naruto. He would have to help Naruto master the Byakugan while still maintaining his existing skills. And Naruto might also have new interactions with other powerful ninjas in the village or from other villages, as they are curious or intimidated by his new - found Byakugan - related abilities.
His friends might be a bit in awe at first. They are used to Naruto's ninja skills, but a Devil Fruit power would be something entirely new. They could also be a bit jealous, but mostly they would be excited to see how Naruto uses this new power in their adventures together.
It might actually strengthen his relationships in some ways. His friends could see him as an even more powerful ally. Naruto, with these new powers, could protect them better. Sakura, for instance, would feel safer around him and this could lead to a deeper bond of trust between them.
It could change his relationship with his enemies too. Enemies might be more hesitant to attack him, not knowing what the mouths on his hands can do. Some might be so freaked out that they avoid him altogether. His relationships with his sensei, like Jiraiya, would be interesting. Jiraiya would likely be excited to study this new phenomenon and help Naruto figure out how to use it to his advantage. And Naruto's new 'hand - mouths' could also bring him closer to some characters who are interested in the strange and unusual in the ninja world.
His relationship with Sasuke would become really complex. Sasuke, as an Uchiha, might feel a mix of jealousy and concern. Jealousy because Naruto now has a part of what makes the Uchiha special, and concern because he knows the dangers of the Sharingan. They might have more intense battles or training sessions as Sasuke tries to understand how Naruto got the Sharingan and how it's affecting him.