Tsunade would probably be very suspicious at first. Naruto suddenly appearing from the future? It's not something that happens every day. She might think it's some kind of enemy's trick or a genjutsu.
Well, Naruto could warn Tsunade about future disasters. Maybe he tells her about Pain's attack on Konoha in advance, so she can better prepare the village defenses. This would change the course of many events and save a lot of lives.
Well, Naruto would probably have a different upbringing. Tsunade could teach him her gambling skills, for better or for worse. He might also be more confident in his abilities from the start, since Tsunade is a very self - assured person. And he could have a different relationship with Jiraiya. Maybe Jiraiya would be more cautious around Naruto, knowing that Tsunade is his mother.
In a fanfiction where Naruto is Tsunade's son, his training regime would be very different. Tsunade would likely push him hard in both physical and medical training. Naruto might have access to resources that he didn't have in the original story. For example, he could study ancient Senju clan jutsu from a young age. His appearance might also be different, perhaps having some of Tsunade's features. His hair color or eye shape could be influenced by her genetics. Moreover, the village's perception of him would change. People would see him as not just the Jinchuriki of the Nine - Tails, but also as Tsunade's heir, which could bring both advantages and disadvantages in terms of how they interact with him.
One idea could be that Naruto inherits Tsunade's incredible strength and medical ninjutsu skills. Maybe in the fanfiction, he uses these skills in a new and unique way during battles, combining his own Rasengan - like techniques with Tsunade - style brute force punches. He could also use the medical ninjutsu to heal his friends more effectively, even in the heat of battle.
The story could also change in terms of Naruto's power development. With Tsunade as his grandmother, he might have access to more advanced medical ninjutsu earlier. This could be crucial in some battles. For example, in the fight against Pain, he could have not only saved the village but also healed many more people quickly. And his understanding of chakra control, which is important for Tsunade's techniques, could make his regular jutsu more powerful too. He could learn to infuse his Rasengan with the same kind of chakra control that Tsunade uses for her super - strong punches.
Tsunade could be a bit flustered at first. After all, she is a tough - as - nails leader, and a hug from Naruto, who she has watched grow up, might make her feel a bit vulnerable in a good way. She might then relax and even give Naruto a pat on the back in return, showing her appreciation for the gesture.
Naruto could also become angry. He might feel betrayed by Tsunade and the village. This anger could fuel his determination to prove his innocence or worth. He could set out on a journey to become even stronger, so that one day he could return and show everyone that they were wrong to banish him. He might also start to question his own beliefs and values, which have been centered around the village and its people.
The other kages from different villages would also have reactions. Gaara, being Naruto's friend, would be happy for him and send a grand gift. The Raikage might be a bit surprised but would respect their decision. The Mizukage would probably see it as an interesting development in the relationships between the ninja villages. All in all, the reactions would range from shock to joy, but in the end, most would support the union.
At first, Naruto would probably be excited. He's always eager to get stronger, so he might see this as an opportunity. He'd trust Tsunade because she's the Hokage and a respected medical ninja. But if the experiments got too painful or started to seem strange, he'd start to question things. He might try to talk to Tsunade about it, but in his Naruto - like way, still be respectful.
He would probably also feel angry. After all, he has sacrificed so much for the village and for them. He might start to question his own beliefs and values. But Naruto being Naruto, he would not give up easily. He would try to prove his innocence and show them that they are wrong. He could go on a journey to find evidence or seek help from other allies who still believe in him, like Shikamaru or Kakashi.