Fynn will reprise his role as Dewey Finn, which he originated in 2016, for the musical's final months.
It was announced earlier this month that School of Rock will close in the West End on March 1, 2020 after more than 1,000 shows.
Fynn will join the company from January 20, taking over the role from Noel Sullivan.
He will lead the adult cast alongside Laura Tebbutt as Rosalie Mullins, Mark Anderson as Ned Schneebly, Rebecca LaChance as Patty Di Marco with Jake Sharp playing Dewey Finn at certain performances.
Based on the movie of the same name, School Of Rock follows Dewey Finn, a failed, wannabe rock star who decides to earn a few extra bucks by posing as a substitute teacher at a prestigious prep school where he turns a class of straight-A students into a guitar-shredding, bass-slapping, mind-blowing rock band.
The hit stage adaption of the 2003 film has been playing at the Gillian Lynne Theatre since 2016.
After closing in the West End, the production will embark on its first UK tour in 2021.
Meanwhile, Andrew Lloyd Webber's Cinderella will open at the Gillian Lynne Theatre from September.
Before then, the theatre will undergo a refurbishment which will include more toilets and a renovated auditorium.