The Queen has said it is “difficult to escape a very sombre national mood” following tragedies in London and Manchester.
Meanwhile, Theresa May has pledged to “get to the bottom” of the west London tower block fire amid mounting criticism of her response to the disaster.
The prime minister was jeered when she visited the Kensington site on Friday, and protests were held in the city.
The death toll is at least 30.
In a statement issued on her official birthday, the Queen added: “Put to the test, the United Kingdom has been resolute in the face of adversity.”
Mrs May was criticised for not speaking to Grenfell Tower residents on her first visit following the blaze.
She told BBC Newsnight the government was doing all it could to help.
The PM said: “I’ve heard stories yesterday from the emergency services about the issues around the fire.
“That’s why I came straight back to Downing Street and I ordered a public inquiry.
“And we’ll make sure that takes place as soon as possible to get to the bottom of this.”
The prime minister has committed £5m for clothes, food and emergency supplies but was heckled with chants of “coward” after she met survivors of the fire.
Asked on Newsnight whether she had misjudged the public mood over the tragedy, she said: “This was a terrible tragedy that took place.
“People have lost their lives and others have lost everything.
“What we are doing is putting in place the support that will help them.”
The BBC understands those missing could number about 70, with the 30 likely to be among that number. Three of those who died have been identified.
Protests were held in London on Friday as residents demanded support for those affected by the fire.
Between 50 and 60 people stormed Kensington and Chelsea Town Hall as members of the public said the homeless needed help “right now”.
There were also angry scenes outside the Clement James Centre, in North Kensington, where the meeting with the prime minister and residents of the tower had been held.
Dozens of demonstrators surged towards the entrance and there were scuffles outside as organisers appealed for calm.
Later, hundreds of mourners stood arm in arm at a vigil and held a two-minute silence for victims of the fire.
Many wept openly as candles illuminated the road outside the Latymer Christian Centre, yards from the site of the blaze.
Earlier, emergency services spent a third day searching for bodies in the burnt-out tower in North Kensington.
The Queen and Duke of Cambridge met volunteers, residents and community representatives during a visit to the Westway Sports Centre.
Asked about the reaction of the crowd, Mrs May defended the government’s response.
“What I am now absolutely focused on is ensuring we get that support on the ground,” she said.
“The government is making money available, we are ensuring we are going to get to the bottom of what has happened, we will ensure that people are re-housed. We need to make sure that actually happens.”
Mrs May had faced criticism for not meeting with survivors in the immediate aftermath, unlike Labour leader Jeremy Corbyn and Mayor of London Sadiq Khan.
Mr Corbyn has written an open letter to the prime minister, calling for the public inquiry to ensure “all necessary lessons are learned”.
The £5m Grenfell Tower Residents’ Discretionary Fund, announced by Mrs May, includes the aim to re-house residents within three weeks as close to where they lived before as possible, to pay for temporary housing in the meantime and to provide extra financial assistance.
So far in the investigation:
- Six victims of the blaze have been provisionally identified by police
- Of those killed, one died in hospital
- Fire chiefs say they do not expect to find more survivors
- A total of 24 people remained in hospital – 12 in a critical condition
- A criminal investigation has been launched
- UK councils are carrying out urgent reviews of their tower blocks, according to the Local Government Association
- The British Red Cross has launched an appeal to raise money for those affected
- The emergency number for people concerned about friends and family is 0800 0961 233
First victims named
On Thursday, the first victim of the fire was named as Syrian refugee Mohammed Alhajali, 23.
The Syria Solidarity Campaign said Mr Alhajali, a civil engineering student, had been in a flat on the 14th floor when the fire broke out, and had spent two hours on the phone to a friend in Syria.
The fire broke out shortly before 01:00 BST on Wednesday.
It tore through all floors of the building and took more than 200 firefighters 24 hours to bring it under control.
There was nothing to suggest the fire was started deliberately, police said.
Two other victims have also been named.
Five-year-old Isaac Shawo reportedly got separated from his family in the smoke and later died.
Artist and photographer Khadija Saye, 24, lived on the 20th floor and also died.