Around 80 firefighters and 12 fire engines tackled the blaze in St Marks Square at just after 9.50am today (August 15).
Part of a three roomed flat on the tenth floor and half of a three roomed flat on the eleventh floor were damaged by the blaze.
Two people were treated at the scene by London Ambulance Service crews and taken to hospital.
Their conditions are unknown.
The brigade has urged residents to avoid the area as significant congestion remains in Westmoreland Road.
Station Commander Gary Melia, who attended the scene, said: “Firefighters worked hard to bring this fire under control.
“There remains significant congestion around Westmoreland Road, so please continue to avoid the area if you can.
“A rest centre has been set up at St Marks Church Hall and we are working with multi-agency partners to support residents.
“Crews will remain on scene for several hours damping down.”
The fire was under control by 11.55am.
Crews from Bromley, Beckenham, Woodside and surrounding fire stations attended the scene.
The cause of the fire is under investigation.
