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
Scotland Yard said the male youths will appear in custody at Westminster Magistrates' Court on Saturday.
The family of Oliver Dearlove urged people to "think again" before using their fists, as the man who killed him began a six-year jail term.
London Fire Brigade took Southwark Council to court on four charges in relation to safety breaches at Lakanal House in Camberwell.