Residents are being asked to avoid the area and keep windows closed.
Surrey Police confirmed Thames Street was closed between South Street and Church Street and that the force was on the scene supporting the fire service.
The fire service also said closures were in place on Riverside Gardens, Clarence Street and Staines bridge.
A spokesman said the fire was now under control but firefighters expected to be on site for “a while longer”.
They said everyone was accounted for, while paramedics assessed four patients for smoke inhalation. One was taken to hospital for further treatment, a South East Coast Ambulance spokesperson said.
They added that its hazardous area response team is at the scene.
