A fire service spokesman said the whole of the roof of the museum and library was on fire.
Station Manager Peter Critchell, at the scene, said:
"Flames are coming through the roof and there is thick grey smoke billowing up. Firefighters are still working hard at the scene to try and bring the blaze under control. We now have water jets attacking the fire from the four corners of the building. We evacuated 30 people from inside the museum and firefighters will be making efforts to salvage as many of the artefacts as possible."
More top news
The footage was apparently shot in south London showing a motorist eating a bowl of cereal, and texting whilst sitting in traffic.
The Prime Minister is trying to curb the money laundering that goes on at the highest ends of the London property market.
The average increase across the capital is 9.2% this year - with areas in the east and south of the city up by 15-16%.