Many are demanding answers for how the blaze spread so quickly. British media have reported that contractors installed a cheaper, less flame-resistant type of exterior paneling on the 24-story tower in a renovation that was completed just previous year.
If at least 58 deaths are confirmed, the blaze would be London's deadliest since World War II.
Sixteen bodies have been taken to the mortuary after flames tore through the 1970s tower block - while 14 others have been recovered from the building.
Police said at least 58 people died, or are missing, presumed dead, after fire ravaged the 24-storey tower block in the early hours of Wednesday.
'Family liaison officers are supporting families, and that includes those people we know to be dead; some of those who are critically ill and sadly those people who we have been told were in Grenfell Tower that night who we have been unable to trace.
"During recent visits in Manchester and London, I have been profoundly struck by the immediate inclination of people throughout the country to offer comfort and support to those in desperate need", the queen said in a statement. She said after the meeting that there have been "huge frustrations" in the community as people tried to get information.
"How many of those have been inspected by fire services and local authorities with maximum transparency?"
Commander Stuart Cundy of the Metropolitan Police warned that the death toll could rise further as he formally identified the first victim as 23-year-old Syrian refugee Mohammad Alhajali. One opposition lawmaker urged the Conservative government and police to immediately seize all documents related to the tower renovation.
Police say the harrowing search for remains had paused Friday because of safety concerns at the blackened tower but has resumed. Some said they had never seen a building fire advance so quickly.
Mr Hammond said: "We've looked at those recommendations and what's happened to them".
The meeting is unlikely to quell complaints that May has been slow to reach out to fire survivors, despite her announcement of a $6.4 million emergency fund to help the displaced families.
"Four hundred or so people - still majority have not got somewhere decent or safe or secure to stay", he said. Protestors have called for her resignation, and she admits that the response should have been better in the initial hours after the 24-floor building was set ablaze.
The identification of the victims is proving very hard - which experts attribute to the extreme heat of the fire.
"Khan went on to call Wednesday's massive Grenfell Tower fire "a preventable accident that didn't need to happen" - and said the tragedy was "a effect of the mistakes and neglect from politicians from the council and from the government".
Police said they are using the INTERPOL Disaster Victim Identification Standards to identify the deceased.
They made images public to give people an idea of the scale of the challenge they are facing as they now try to identify victims.
The queen and Philip stood silently before the start of the annual Trooping the Color procession that marks the queen's official birthday.