All Might would be furious. He would see it as his failure to protect Izuku. He would probably mobilize all the resources he has, both in his hero form and his connections in the hero world, to rescue Izuku. And he would be blaming himself all the while for not being there when Izuku was kidnapped.
The villains in the story would either be gloating about their success in kidnapping Izuku or be on edge. If they kidnapped him for a specific purpose, like using his knowledge of hero strategies, they would be eager to start interrogating him. But if they knew that Izuku's friends and the heroes would be coming after them, they would be preparing for a fight.
Izuku would probably be terrified at first. After all, he's been kidnapped by villains. But then, his hero - like determination would kick in. He might start looking for ways to escape or at least resist their plans.
Some characters might be immediately terrified. They could avoid Izuku or even try to drive him away. For example, classmates who don't understand his quirk might spread rumors about him.
All Might could also be a potential hero to save Izuku. He has a strong bond with Izuku and his presence alone could intimidate Overhaul. He would use his experience and power to rescue Izuku from the dangerous situation.
Maybe it's a group of villains working together. They could be from different factions but united by the idea that Izuku has the potential to become a huge threat to their plans. They might kidnap him to either eliminate him or turn him to their side. Also, it could be a lone, obsessed fan of a villain who wants to prove themselves by bringing Izuku to their idol.
Often, Izuku reacts with shock and disbelief at first. He can't fathom why someone he trusted would turn on him.
Hiro would be determined to save the kidnapped member. He'd immediately start analyzing the situation, using his intelligence to figure out where they might be held. Baymax would be concerned for the safety of the kidnapped character and would follow Hiro's lead. GoGo would be angry and ready to take action, using her speed to quickly track down any leads. Wasabi would be a bit more cautious but still committed to the rescue. Fred would probably be coming up with wild but potentially useful ideas on how to save their friend.
There could be his friends like Ochaco Uraraka. She might be in a position where she's trying to understand Izuku's situation and help him. Maybe she tries to convince the people forcing Izuku to give him some space.
Bakugo is often paired with Izuku. Their relationship is complex in the original series, and in fanfiction with Izuku's time quirk, it could add more depth. Bakugo might be jealous of Izuku's new power at first but then come to respect it as they face challenges together.
Bakugo is often paired with Izuku. Their complex relationship from the original series can be further explored in fanfiction with Izuku's growth quirk. All Might is also a common character paired with Izuku. He could play a role in guiding Izuku to control and develop his growth quirk. Ochaco is another one. She might be involved in stories where Izuku's growth quirk affects his relationships, especially romantic ones.
All Might would probably be in it. Since he's Izuku's mentor, he could play a role in helping Izuku understand and control his time - controlling power. Maybe he has some past experience related to time powers that he shares with Izuku.