Commander Stuart Cundy said they had been able to remove the remains of only 12 victims from the building.
The official death toll from the blaze now sits at 30, up from 17 overnight, as firefighters continue the extremely hard process of recovering bodies from within the tower.
The prime minister is still reeling from a botched snap election that resulted in her Conservative Party's loss of its majority in Parliament.
Criticism of May intensified Friday after she sidestepped questions in a televised interview about whether she had underestimated the public's anger and frustration.More news: Representative Scalise's condition upgraded to 'serious' after shooting
As the video continues to record, Rania is heard praying and saying police are telling people to get out but that they can't due to the fire and the smoke.
"But we may well need help from our close neighbours".
He told reporters that the investigation's "absolute priority" is to establish the identities of those inside the building.
Mrs May has pledged a £5 million support package for the victims, which included a promise to rehouse them within three weeks. "People have lost their lives, people have had their homes destroyed, they have fled for their lives with absolutely nothing".
As the victims of the fire entered the Downing Street, hundreds of demonstrators had gathered in Whitehall to protest against the prime minister.
"The Prime Minister would have been shouted at by the residents, but she should have been willing to take that".More news: Cosby jury ends first day of deliberations without a verdict
"I know that they are looking at us to save them".
Two boys inspect pictures of the missing.
Similarly, the mother of Hesham Rahman who also lived in the Grenfell Tower said that her son remains missing.
A group of angry protestors on Friday had entered the Kensington Town Hall shouting "we want justice" and "shame on you", according to Reuters.
The 91-year-old monarch said it is "difficult to escape a very sombre mood" on what is normally a day of celebration.
He said: "Little can compare in recent times to a tragedy on this scale - the response from the local authority has simply not been good enough".More news: 58 presumed dead in London tower blaze, police say
"Put to the test, the United Kingdom has been resolute in the face of adversity", Elizabeth said.
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