The fire service was called to the restaurant on High Street at 2.41pm.
Ducting from the first floor to the roof, part of the first floor and part of the roof were damaged by the fire.
There were no injuries although the London Ambulance Service attended the scene.
The police are “holding the affected flats as a crime scene” according to a local councillor.
60 firefighters from Sutton, Wallington, Mitcham, Wimbledon, New Malden and surrounding fire stations brought the fire under control by 8.45pm.
The London Fire Brigade will carry out an investigation into the cause of the fire.
Cllr Choi said during the fire: “Police, Fire and Ambulance services are currently on scene dealing with a fire within the flats above Wendy’s.
“A large section of the High Street in front of Wendy’s has been cordoned off to enable the emergency services to respond to the incident and ensure the safety of the public.
“We can confirm that no residents have been required to evacuate, and thankfully, there have been no injuries reported to any members of the public.
“The police are currently holding the affected flats as a crime scene, and a thorough investigation will follow to establish the cause of the fire.”
