A mission against the Kraang gone wrong to some *ahem* mishaps caused by Mikey, the turtles, Splinter, the Kraang and even Shredder and some of his grunts get transported to the MHA world, though the brothers are together, everyone else is scattered. Sticking with what they know best, they lay low and gather information about the new world they're on, while also finding refuge in an abandoned subway station.
While being trained as ninja since they were young, only Leo spoke fluent Japanese, with Donnie being the second most fluent just not as fluent as Leo, and Raph and Mikey being last. So they often stayed in a group but if they had to split, they'd be in teams of two, while looking for Splinter and keeping an eye out for their enemies.
Things get harder when they have heroes on their tails claiming they should turn themselves in for illegal quirk use (the heroes think their ninja skills attribute to quirks).
Through some connections they gain later on, and clarification, they're able to continue their search as long as they have a hero with them and attend UA for observation.
I'm not sure what to have Shredder do while in this world, so I guess that's up to you.