The fire at Grenfell Tower on the Lancaster West Estate in Latimer Road was reported at 01:16 local time. About 600 people were believed to have been inside the tower's 120 flats, many of them asleep, when the blaze ripped through the building.
Firefighters rescued many people but there may still be people in the building who are unaccounted for.
Many people from neighbouring housing blocks have also been evacuated as a precaution and gathered at community centres, mosques, gurdwaras and churches in the area until the structural credibility of the burning tower is verified.
Vinod Diwakar, the Indian-origin medical director of NHS England's London region, whose team has been coordinating the medical response, urged Londoners to try and ease the pressure on the NHS as medical staff deal with the tragedy.
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"The hospitals receiving patients continue to be busier than normal and we encourage Londoners to use NHS services wisely and turn first of all to sources of advice such as NHS 111 and local pharmacies," he said.
Some reports have suggested that the building's rain- screen cladding, installed as part of a 10-million-pound refurbishment work last year may have contributed to the speed with which the fire spread through the entire building.
Metropolitan Police Commander Stuart Cundy said the force do not expect to find any more survivors. "This is going to be a long and complex recovery operation and I do anticipate that the number of fatalities will sadly increase beyond those 12," he said.
Steve Apter, director of safety and assurance at London Fire Brigade, said the inferno was "unprecedented in its scale and speed" and firefighters have now "searched almost all the building" and managed to get to the top-most floor.
"There are still pockets of fire yet to be extinguished in particularly difficult areas to reach," he said, adding his officers will carry on their work on the site overnight.
London Fire Brigade chief Dany Cotton said: "This is an unprecedented incident. In my 29 years of being a firefighter, I have never ever seen anything of this scale."
Eyewitnesses described people trapped in the burning tower, screaming for help and yelling for their children to be saved. They said they saw lights - thought to be mobile phones or torches - flashing at the top of the block of flats, and trapped residents coming to their windows - some holding children.
Some residents were seen using bedsheets to make their escape from the tower block.
"I'm lucky to be alive - and lots of people have not got out of the building I've lost everything I own. I'm standing here in everything I've got," one survivor said.
"We can't have a situation where people's safety is put at risk because of bad advice being given or if it is the case, as has been alleged, of tower blocks not being properly serviced or maintained," he said.
Fire Commissioner Dany Cotton said firefighters expected to be on the scene for at least another 24 hours and she would not speculate about the cause of the blaze.
She said there were concerns that people were still inside the tower and she urged all residents to make sure they had reported themselves to police.
The area around Grenfell Tower is home to a large number of Muslims. Many were awake at the time the fire broke out having their early morning meal before beginning the daily fast for the holy month of Ramzan.
The survivors, whose belongings are presumed to have been destroyed, gathered in the nearby Rugby Portobello community centre where they were given water, clothes and blankets.
Michael Paramasivan, who lives on the seventh floor with his girlfriend and young daughter, said he ignored official advice to stay in your home.
Paul Munakr, who lives on the seventh floor, spoke of his escape. "As I was going down the stairs, there were firefighters, truly amazing firefighters that were actually going upstairs, to the fire, trying to get as many people out the building as possible," he told the BBC.
Another resident, Zoe, who lives on the fourth floor, said she was woken by a neighbour banging on her door.
Refurbishment of the housing block had been completed last year and the Fire Brigades Union said something had gone badly wrong with fire prevention procedures at the building.
London Mayor Khan declared the fire as a "major incident" and said questions will need to be answered over the safety of such tower blocks.
"We can't have a situation where people's safety is put at risk because of bad advice being given or if it is the case, as has been alleged, of tower blocks not being properly serviced or maintained," he said.
The local Grenfell Action Grouphad claimed, before and during the refurbishment, that the block constituted a fire risk and residents had warned that access to the site for emergency vehicles was "severely restricted".
In a statement, Kensington and Chelsea Tenant Management Organisation acknowledged concerns had previously been raised.
"We are aware that concerns have been raised historically by residents. We always take all concerns seriously and these will form part of our forthcoming investigations," it said. "We will cooperate fully with all the relevant authorities in order to ascertain the cause of this tragedy," it added.