He would likely have enhanced physical strength and speed, similar to Saiyans. Maybe he could fly like them too. His combat skills would be more brutal and direct, rather than the ninja - style he has in the original. He might also develop some sort of energy blast ability like Vegeta's Galick Gun.
Naruto would have some really interesting powers. For one, his physical abilities would be super - enhanced. He'd be able to jump higher, run faster, and hit harder. His power - up techniques would be more in line with Saiyan transformations. He could potentially have a form that gives him a boost in power similar to Super Saiyan. His fighting style would incorporate more energy - based attacks. And with Vegeta's training, he'd be able to sense and manipulate energy in ways that are foreign to the traditional Naruto skills. This would make him a formidable opponent, with a unique blend of Naruto's natural potential and Saiyan - inspired powers.
In this fanfiction scenario, Naruto would inherit a host of Saiyan powers on top of his ninja skills. For starters, he would have the Saiyan's natural fighting instincts and the ability to quickly recover from injuries. His power level would constantly increase as he fought, similar to how Saiyans get stronger through battles. His existing jutsus like the Rasenshuriken could be infused with Saiyan energy, making it not only cut through opponents but also blast them with a powerful ki - like force. And with his Saiyan tail, if he had one, he might have an additional form or power that was unique to him, perhaps something that combined the power of the Great Ape form with his Nine - Tailed Fox form.
His powers could be a mix of ninja skills and Thanos - like abilities. He might have enhanced physical strength similar to Thanos. His Rasengan could potentially be charged with some sort of energy related to the Infinity Stones. Also, his ability to sense chakra might be modified to sense other forms of power. Maybe he could even teleport like Thanos does in the comics, which would give him a huge advantage in combat.
In a 'Gohan Raised by Vegeta' fanfiction, Gohan's personality would likely change a great deal. Vegeta being his mentor would mean a strict and no - nonsense upbringing. Gohan would probably be more confident in his Saiyan heritage. His fighting style could be a blend of Vegeta's elegance in combat and his own natural potential. Also, the relationship between Gohan and other characters like Piccolo or Chi - Chi would be very different. With Vegeta's influence, he might be more rebellious against Chi - Chi's wishes for him to focus on academics.
If Naruto was raised by Vegeta, his character would likely become much more aggressive and focused on power. Vegeta's Saiyan pride and his constant drive to be the strongest would rub off on Naruto. He might not be as friendly or as empathetic as in the original series, but would instead be more solitary in his pursuit of strength, seeing it as the ultimate goal.
His chakra control might improve in a different way. Sesshomaru's precise and refined combat style could force Naruto to have better control over his chakra. For example, he might learn to use his chakra more sparingly like Sesshomaru uses his demonic energy.
Naruto's powers would likely be a blend. His existing ninja abilities like shadow clones might be adapted to Mandalorian combat scenarios. He could use his clones to create diversions while he attacks with Mandalorian weapons. Also, the Mandalorian training in stealth and marksmanship could make his ranged attacks more effective, adding a new dimension to his power set.
Well, it would be really strange and interesting. Vegeta is from Dragon Ball, a very different universe from Harry Potter's. If Harry was raised by Vegeta, he might be trained in combat from a very young age, like how Vegeta trains Trunks. He could be more aggressive and less reliant on magic wands, perhaps using his ki powers instead. And his attitude towards enemies would be more like Vegeta's, full of pride and determination.
If Naruto was born in the Uchiha clan in fanfiction, he would surely have the Sharingan. This powerful eye technique would give him enhanced perception, allowing him to copy other ninja's moves more easily. He could also use Genjutsu more effectively. Along with that, he might still have some connection to the Nine - Tails, but it could be more subdued or combined with his Uchiha powers in a unique way. Maybe he could use the Nine - Tails' chakra to boost his Sharingan abilities, like increasing the power of his Amaterasu if he were to unlock it.
He might have powers like shape - shifting into a more demonic form. Maybe he could have the ability to breathe fire or some other elemental powers related to chaos or destruction. He could also have enhanced strength and speed that are far beyond what a normal ninja has, and his senses could be super - sharp to detect danger or prey.
He might have the power of instant teleportation. Just like a god can appear anywhere, Naruto could do the same to surprise his enemies or reach his friends in need quickly.