Ten people fled from the building in Grenfell Road when the fire started around 8.30pm yesterday (April 6).
The stairwell filled with smoke, meaning a further three people had to be saved by the fire service through a window using a ladder.
A woman and man were treated by the ambulance service and taken to hospital along with the baby boy and girl.
The fire was extinguished in under two hours at around 10.20pm by 25 firefighters from Tooting, Mitcham and Wimbledon fire stations.
The fire is thought to have been accidentally started by a candle, sparking a fresh warning from a fire station officer who was at the scene.
Edmund Gonzales said: “Candles are one of the most common causes of fire.
“It’s really important that you never leave a candle unattended and keep them away from anything else that can easily catch alight.
“Consider using battery-powered, flameless LED candles instead, which are safe even if you fall asleep.
“Whilst they may cost more upfront than traditional candles, they will last considerably longer, giving you better value for money.
“If you do choose to use traditional candles ensure you place them in a heat-resistant holder, on a stable surface where they won’t be knocked over and away from materials that could catch fire.
“Never leave them unattended around your home, take particular care with children and pets and blow them out properly before you go to bed.
“This incident shows just how important it is to have working smoke alarms fitted on every level of your home.
“Smoke alarms give the earliest possible warning when a fire starts and we would urge everyone to make sure they have one fitted in every room where a fire can start except kitchens or bathrooms where heat alarms are more appropriate.
“We also recommend taking some time to carry out a few simple checks to keep yourself and your loved ones safe.
“The Brigade’s online Home Fire Safety Checker can guide you around your home and provide you with specific advice to reduce fire risks.”
