She also noted that an emergency fund of more than $6 million has begun to be "distributed on the ground so people can buy clothes, food and other essentials".
London is reeling from Wednesday's early morning blaze, which charred the bulk of the 120-apartment tower in the city's Kensington neighborhood.
An additional £1.5 million will be provided to pay for mental health support to the emergency services, through mental health charity Mind's Blue Light Programme.
"We've worked tirelessly to confirm the number of people who we believe were in the tower on the night and, at this time we can not assume they are safe and well", he said.
Protestors at Kensington town hall were trying to reach councillors in person and demand they answer questions about the block.
The protesters barged their way through an automatic door and sought to gain entry to an upper floor.
Told there was a need for the public to hear her say something had gone badly wrong and the Government accepted responsibility, Mrs May said: "Something awful has happened".
Almost 200 people chanting "We want justice!" marched near Notting Hill Gate tube station on Friday, demanding action from local councilors tasked with housing people following the disaster.
Theresa May has admitted that the support given to families in the hours following the Grenfell Tower fire "was not good enough". Demonstrators gathered outside, protesting about several issues including the fire.
Protesters sit down in Oxford Circus during a rally calling for justice for those affected by the Grenfell Tower fire on June 16. Instead they released a statement promising to rehouse as many people locally as they could.
Public anger is mounting as residents and neighbors demand answers for how the blaze early Wednesday spread so quickly and trapped so numerous tower's 600-odd residents.
Cundy said the police investigation would look into the building's 2016 refurbishment and promised to prosecute "if there is evidence".
The criticism over the blaze has added to pressure on May who is trying to strike a deal with a Northern Irish party to stay in power.
May said she would be responsible for implementing the inquiry's findings, while the victims' legal costs would be met by the government.
"We've worked tirelessly to establish how many people we believe were in Grenfell Tower on the night and at this point in time we are unable to say that they are safe or well", he said. "They have been left with nothing - no bank cards, no money, no means of caring for their children or relatives". Many chased after her armored cars, prompting the police to intervene and push people back.
Communities and Local Government Secretary Sajid Javid said inspections of similar buildings had been ordered, with particular attention to the modern cladding used to beautify and add insulation to ageing concrete and steel structures.
"She should have been there with the residents".
London Mayor Sadiq Khan said Saturday he relayed residents' multiple concerns to a taskforce assembled in response to the fatal blaze. A memorial wall has been created for people to write messages to the missing. Residents complained of problems with the building, which they said mostly went unanswered. Nineteen people are still in the hospital, 10 of whom are in critical condition.
However, Foreign Secretary Boris Johnson, who was London mayor until a year ago, took to Facebook to attack opposition Labour politicians for "political game playing". May was jeered on Friday after she visited the affected community.
The queen visited a shelter for victims earlier on Friday to speak with displaced residents.
Cundy said the search was paused over safety concerns Friday, but has since resumed. "Sadly, we do not expect any survivors".