Mr Cundy said the search-and-recovery operation was ongoing in the burnt-out, 24-storey tower.
Cundy said five people who had been reported as missing had now been found and were safe and well.
Five people have been formally identified, including 23-year-old Mohammad Alhajali.
A minute's silence was held at 11:00 BST on Monday across the United Kingdom to remember the victims.
He said: "This is incredibly distressing time for families".
Commander Stuart Cundy, of the Metropolitan Police Service (MPS), said another 26 people are now presumed to have died in the fire, raising the total number to 79.
Firefighters told the BBC programme that they had seen flames "shooting up" the side of the building.
He said it had been "incredibly emotional working in there", adding: "On Saturday I went in myself and went to the top floor".
Grenfell Tower residents condemn 'invisible' estate managers
The queen's official birthday is marked in June when the weather is often nicer than in April, the actual month of her birth. At 58 casualties, it would make the Grenfell Tower blaze the deadliest in the capital since the Second World War.
However, Mr Cundy went on to describe the search operation as "hugely complex" and said: "The terrible reality is, as I've said before, is that due to the intensity of the fire and the devastation within Grenfell Tower, we may not be able to identify everybody that died".
The GRT, which includes local and regional government from across London, central government, the British Red Cross, Met Police and London Fire Brigade, said in a statement that "we completely accept residents' wishes to remain close to the community".
"The prime minister is very aware that people want answers promptly and we want to get this going promptly, so that's our intention", the spokeswoman said.
Speaking outside Scotland Yard, Mr Cundy added: "I have investigated major crime for most of my service and I have seen some awful things.
She understood that immediate action needed to be taken to speed things up, and that's what she's done", the spokeswoman said.
The concerns were raised by Councillor Anthony Mckeown who asked the council to clarify about the insulation works that were completed on Gamesley and Fairfield a few years ago.
Prince William paid a visit to a crisis centre on Friday to show support to the victims of the deadly West London blaze, which had a death toll that rose to 58 victims. I should point out that assessments carried out to comply with the Fire Safety Order are unlikely to have considered the building's cladding, ' Dawes added.