The Queen has said it is "difficult to escape a very sombre national mood" following tragedies in London and Manchester in recent weeks.
The monarch said the UK had "witnessed a succession of terrible tragedies"
Her message comes as residents caught up in the Grenfell Tower fire which killed at least 30 people, condemned the "chaotic" relief effort.
Some say they no longer want Kensington and Chelsea council involved in any way.
'United in our sadness'
The statement from the Queen on her official birthday came after Prime Minister Theresa May pledged to "get to the bottom" of the west London tower block fire.
Mrs May was jeered when she visited the North Kensington estate on Friday, and is facing mounting criticism for her response to the disaster.
She is now meeting victims, volunteers and community leaders in Downing Street, while hundreds of protesters have gathered in Whitehall, calling for her resignation.
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The Queen and Duke of Cambridge had earlier met volunteers, residents and community representatives during a visit to the Westway Sports Centre.
She also visited some of those injured in hospital following the bomb at Manchester Arena little under a month ago.
In an unprecedented statement, the Queen said she had been "profoundly struck by the immediate inclination of people throughout the country to offer comfort and support to those in desperate need".
"Put to the test, the United Kingdom has been resolute in the face of adversity," she said.
"United in our sadness, we are equally determined, without fear or favour, to support all those rebuilding lives so horribly affected by injury and loss."
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A rallying cry for unity
By Peter Hunt, BBC royal correspondent
In a long reign, the Queen has issued countless statements. They can sometimes be formulaic and lack the ability to resonate.
This is not one of them.
After the attacks in Manchester and London, the Queen – and more importantly her senior advisers – have grasped quickly that the reaction to the Grenfell Tower fire has not just been one of shock and grief.
There's also intense anger. It has been focused on the divide between rich and poor and it has been directed at an establishment that includes the monarchy, though the institution itself has not been the subject of criticism.
In such circumstances, as Head of Nation – a focal point at times of tragedy – the Queen has decided she cannot remain silent.
She, and those around her, will be acutely aware of the potential for growing disquiet in the days ahead.
And so, a 91-year-old monarch with little concrete power but considerable patronage and status, has decided to act.
This is a regal rallying call for unity.
The fire broke out at the 24-storey block, which contained 120 one and two-bedroom flats, 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.
Cladding installed on the tower during a refurbishment in 2015 had a polyethylene – or plastic – core instead of an even more fireproof alternative, the BBC's Newsnight understands.
The Department for Communities and Local Government said that cladding made up of aluminium panels over a polyethylene core "would be non-compliant with current Building Regulations guidance" and should not be used on buildings taller than 18m.
It said it could not comment on the type of cladding used on Grenfell Tower. That would be subject to investigations, it said.
Protests were held in London on Friday as residents demanded more 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.
Dozens of demonstrators surged towards the entrance and scuffles broke out outside as organisers appealed for calm.
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The BBC's Matthew Price said locals had told him they wanted the prime minister to remove Kensington and Chelsea council from the relief effort.
He said senior members of the residents' association described an "absolute chaos" of "no organisation" from officials.
He added: "They do not believe they are capable of managing the response. Such is the total and utter lack of trust."
Reverend Mike Long, from Notting Hill Methodist Church, told BBC Radio 4's Today that people in the community were furious.
"People are incredibly angry, they're bewildered, they're confused, they have lots and lots of questions," he said.
"They feel they're not being listened to and what they have been saying has not been listened to, and they don't know how to be able to express those things at the moment."
Mrs May has faced criticism for not meeting survivors in the immediate aftermath, unlike Labour leader Jeremy Corbyn and Mayor of London Sadiq Khan.
Mr Corbyn has called for empty local homes to be requisitioned if necessary.
The prime minister has committed £5m for clothes, food and emergency supplies but was heckled with chants of "coward" after meeting survivors of the fire on Friday.
First Secretary of State Damian Green has defended her, saying the criticism was "unfair" and she was "as distraught as all of us".
The £5m Grenfell Tower Residents' Discretionary Fund 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.
On Friday night, 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.
It came after emergency services spent a third day searching for bodies in the burnt-out tower in North Kensington.
In latest developments:
- A minute's silence was observed by the Queen at the Trooping the Colour parade to remember the victims
- Mrs May's new taskforce, made up of central government and Kensington and Chelsea council representatives, is due to meet
- Cardinal Vincent Nichols, the Catholic Archbishop of Westminster, is leading an adoration and mass in memory of the victims at St Pius X Church
- The Circle line and Hammersmith and City underground lines, which run close to the tower, are partly suspended at the request of fire chiefs following a "short-term risk of some debris falling onto the tracks"
The Confederation of Fire Protection Associations-International – a global body of fire protection organisations – said the speed at which the fire spread at Grenfell Tower was "eerily similar" to other high-rise fires in Australia and the United Arab Emirates.
It said it was "deeply concerned" that there were many high-rise buildings around the world with flammable materials installed.
So far in the investigation:
- Six victims of the blaze have been provisionally identified by police
- Three have been named so far, including Syrian refugee Mohammed Alhajali, 23., five-year-old Isaac Shawo, and artist Khadija Saye
- 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