Without saying too much, I think their personalities are true to their TOS selves, until the caricatures the recasts were in the Abrams movies.

Nurse Chapel is the only classic character that’s been completely reimagined. She’s more more active, and motivated. She’s still got a thing for Spock, but she isn’t just some hopeless wilting flower. This does clash with TOS, but I accept it as overriding TOS Chapel.

Dr. M’Benga, who is a very obscure reoccurring TOS character is the ship’s doctor, and as is Trek tradition, has a lot of screen time.