One consequence could be that he faces opposition from some clan members who might not be ready to accept him as the leader right away. After all, taking over a clan is a big deal.
In some fanfics, Naruto might inherit the powerful chakra control abilities typical of his Uzumaki clan. This could allow him to perform complex jutsu with greater ease compared to the canon Naruto.
It could be due to a prophecy. The Uzumaki clan might have received a prophecy that Naruto would bring great turmoil or change to the clan, and rather than face the uncertainty, they chose to exile him. Some clans in the Naruto universe are very superstitious about prophecies. Also, Naruto's unruly nature as a child could have been misinterpreted. His pranks and disregard for some of the strict clan traditions might have been seen as disrespectful and a sign that he didn't belong in the Uzumaki clan, leading to his exile.
One idea could be a story where Naruto and Hinata explore the hidden secrets of the Uzumaki clan together. They might discover ancient jutsu or artifacts that could change the future of the Hidden Leaf. For example, they could find a powerful sealing technique that was lost for centuries and use it to protect the village from a new threat.
Maybe Naruto proves himself worthy by showing off his powerful jutsu skills that are related to the Uzumaki heritage. He could also use his status as a hero in the village. By winning the trust and respect of the Uzumaki clan members, he gradually takes over the leadership of the clan.
In fanfiction, Naruto being from a clan often means he has a different heritage. For example, some fanfics might make him part of an ancient and powerful Uchiha - like clan. This could give him access to new jutsu or abilities that he doesn't have in the original story. It can also change his relationships with other characters. For instance, if he's from a clan that has a feud with the Hyuga, his interactions with Hinata would be more complex.
One challenge would be getting other shinobi to join. Some might be loyal to their existing clans. Also, there could be political opposition in Konoha. The elders might not be too keen on a new clan being formed.
If Naruto saved the Uchiha clan, the story of the Hidden Leaf Village would be rewritten. The history of the Uchiha's massacre would no longer be a dark stain on the village's past. The Uchiha would contribute their unique skills and abilities to the village's development. For example, their Sharingan abilities could be used for better security or in battles against common enemies. There would also be a change in the relationships between characters. People who were once enemies of the Uchiha might have to learn to work with them.
Well, if a Senju became the clan head in fanfiction, it could change a lot of things. For one, they could have a different approach to dealing with the Uchiha. Instead of the historical tensions, they might strive for a more harmonious co - existence. Also, they could play a big role in the overall politics of the Hidden Leaf. They could use their influence to improve the village's infrastructure and ninja training programs.
The Hyuga clan could also be a possibility. Given their strong bloodline and their focus on family and tradition, they might adopt Naruto. They could teach him their Gentle Fist technique. And Naruto could use his unique energy and determination to perhaps improve or modify the technique in some way. Plus, he could help the Hyuga clan with any external threats they face due to their valuable Byakugan.