Transport for London is wholly owned by the Greater London Authority, which is run by the Mayor of London. And since the Mayor is elected, that means TfL is technically publicly owned—just not by the national government in Westminster. It’s like the difference between your rich uncle owning your house, and your weird neighbour saying they “own the street.”
In 2000, Ken Livingstone (the guy who looked like a grumpy frog in a suit) created TfL to bring all of London’s transport under one roof. Before that, it was chaos—like seven different clowns trying to drive the same clown car.