The decision comes after a row over funding shortfall from UK’s Department of Culture, Media and Sport.
Rogers Stirk Harbour will build the British Museum extension, while the Tate Modern extension will be designed by Herzog & de Meuron.
The core of the British Museum Quadrangle building, comprising the four main wings, was designed in 1852 by Sir Robert Smirke.
Smirke designed the building in the Greek Revival style, with features including the columns and pediment at the south entrance.
The Tate Modern houses the UK’s national collection of international modern art and has been converted by Herzog & de Meuron into a museum of modern art, following its former life as a power station.
Tate Modern’s Swiss Light feature and penthouse lightbeam mark the building for many miles.
Under the funding programme, BFI will also receive go-ahead for its new £166m national film centre on the South Bank with the aid of a £45m grant.
The BFI project will have five digital screens and a large auditorium for hosting events, which currently held at the Odeon Leicester Square. It is to be built by 2015.