Police say the harrowing search for remains had paused Friday because of safety concerns at the blackened tower but has resumed.
Mr Cundy said: "The investigation will be exhaustive". He says "my heart goes out to those affected".
Later on Sunday, former United Kingdom chancellor Philip Hammond said the cladding material used on the building was banned on British high-rises.
The announcement comes as the government is accused of failing to improve fire safety in buildings, like Grenfell Tower, despite warnings. A new sign was put up, removing that detail.
Mrs May said: "The inquiry will be open and transparent".
At least 30 people were killed in Wednesday's inferno, which left Grenfell Tower a charred hulk.
Two nearby Underground lines were partially shut down Saturday in the fire area to make sure that debris did not land on the tracks.
Labour Party lawmaker David Lammy said that the government and the police should immediately seize all documents relating to the building's renovation to prevent the destruction of evidence that could show criminal wrongdoing.
Meanwhile, embattled May found herself in more trouble on Sunday, after her administration walked back a promise to house the survivors of the fire in the same neighborhood.
Tailored bereavement support for families of the victims, those missing or people affected by the tragedy will also be provided.
She sought to combat some of the criticism with a meeting with residents, victims, volunteers and community leaders at Downing Street today. Spain faces Macedonia on Saturday night in the tournament, which is being played in Poland.
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Members of the group praised Hodgkinson online after learning about the political assassination attempt. In March, Hodgkinson wrote about his support for a petition to investigate Donald Trump for treason.
In her traditional birthday message, the 91-year-old monarch made a reference to the "succession of awful tragedies", a day after she paid a visit to people affected by the massive Grenfell Tower blaze in west London.
Dozens are still missing.
The queen and her husband, Prince Philip, stood silently on the steps of Buckingham Palace before the start of the Trooping the Color that each year marks the queen's official birthday, which is traditionally celebrated in June when the weather is warm. "It should define the leadership of May and she should be forced to resign", he said.
"We've always been advocates of automatic fire sprinkler protection in tall tower blocks and they work - people don't die in sprinkler buildings", he said.
Families of the victims and residents of Grenfell said they were furious that the local government had for years ignored their complaints about the building's lack of measures in case of fire and had not adequately attended to survivors of the blaze. 24 people remain in hospital, with 12 of those people in critical condition.
In addition, British health authorities will provide long-term bereavement counselling for those who lost loved ones in the tragedy.
"Little can compare in recent times to a tragedy on this scale - the response from the local authority has simply not been good enough". Although the police and fire service expect the number to rise significantly, they are following their usual procedure of only raising it as they confirm deaths.
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".
Mohammed, whose apartment is opposite the tower, had friends who lived in Grenfell's upper stories.
Many are demanding answers for how the blaze spread so quickly.
Britain's Press Association reported that some 70 people are still missing after the fire. "This is an absolutely bad fire that took place", May said. This year, however, it is very hard to escape a very sombre national mood.
The government has promised a full public inquiry.