So far, 58 people have been feared dead - a toll that authorities warn could rise ahead of new figures to be released on Monday.
On the lack of sprinklers in Grenfell Tower, and other buildings, he said: "My understanding is that the best expert advice is that retrofitting sprinklers may not always be the best technical way of ensuring fire safety in a building".
"All I can say is to my certain knowledge the council, councillors, volunteers, community groups, officers of the council, have been present in North Kensington since very soon after they were notified that the fire had started".
Flames engulfed the building in less than an hour early on June 14, trapping many residents before firefighters could reach them.
One of the burned-out apartments inside London's Grenfell Tower.
"The bad reality, as I've said before, is that due to the intensity of the fire and the devastation within Grenfell Tower that we may not be able to identify everybody that died", Cundy told reporters.
Two British ministers say the cladding used in a renovation of Grenfell Tower may have been banned under United Kingdom building regulations.
Ya-Haddy Sisi Saye, also known as Khadija Saye, 24, Abufars Ibrahim, 39, and Anthony Disson, 65, all lived in the 24-storey block, which was destroyed by a huge blaze last Wednesday in west London.
Corbyn said on Sunday that the government has the power to take over empty apartments and offer them to the hundreds of Grenfell Tower residents left homeless by Wednesday's fire.
"The terrible reality is that, due to the intensity of the fire and the devastation within Grenfell Tower, we may not be able to identify everyone who died", he added.
British Prime Minister Theresa May is facing mounting pressure to resign due to her handling of the situation and tensions have been heightened by her failure to meet with victims in the immediate aftermathof the tragedy.
Cundy said a criminal investigation began the day of the fire and will look at a "wide range" of issues, including the renovation project, the public housing block's management and maintenance, and fire-safety measures.
The queen marked her official birthday Saturday by saying Britain remains "resolute in the face of adversity" after the horrendous fire and recent extremist attacks in London and Manchester.
In addition, British health authorities will provide long-term bereavement counselling for those who lost loved ones in the tragedy.
May also said that a public inquiry that is looking into the Grenfell Fire will report directly to her and that she has asked for daily updates from the neighbourhood to be sent to her.