The Queen has sent a message to the US president, Joe Biden, to commemorate the 20th anniversary of 9/11, and remembered her visit to Ground Zero, the site of the attack on the World Trade Center.

Her message said: “As we mark the 20th anniversary of the terrible attacks on September 11 2001, my thoughts and prayers – and those of my family and the entire nation – remain with the victims, survivors and families affected, as well as the first responders and rescue workers called to duty.

“My visit to the site of the World Trade Center in 2010 is held fast in my memory. It reminds me that as we honour those from many nations, faiths and backgrounds who lost their lives, we also pay tribute to the resilience and determination of the communities who joined together to rebuild.”

The British prime minister Boris Johnson said the 9/11 terrorists had failed to undermine the faith of “free peoples” around the world in open societies, in a video message that is to be played during a memorial event at the Olympic Park in east London on Saturday.

The prime minister, who was born in New York City, said the threat of terrorism remained but people refused to live their lives in “permanent fear”.

“The fact that we are coming together today – in sorrow but also in faith and resolve – demonstrates the failure of terrorism and the strength of the bonds between us,” Johnson said.

A total of 2,977 people were killed in the terror attack, including 67 Britons – “each of them a symbol of the eternal friendship between the United Kingdom and the United States,” he said.

The date of 11 September 2001, “became, in President Roosevelt’s words after Pearl Harbor, a ‘date which will live in infamy'”, Johnson added.

“But while the terrorists imposed their burden of grief and suffering, and while the threat persists today, we can now say with the perspective of 20 years that they failed to shake our belief in freedom and democracy. They failed to drive our nations apart, or cause us to abandon our values, or to live in permanent fear,” he said, adding that recent events in Afghanistan had only strengthened people’s belief in freedom and democracy.

The Labour leader, Keir Starmer, said the consequences of the attacks were “still being felt to this day”, adding the tragedy was “still so raw”.

He said: “But as we mark this anniversary I’m convinced our resolve has never been stronger. We will continue to fight terror and violence, by promoting our values of justice and peace.”

The archbishop of Canterbury said on Twitter: “Twenty years after the 9/11 attacks, it’s still hard to articulate the loss that terrible day brought.

“Today we pray for all those affected by terror and violence. May God guide us together towards the peace offered by the crucified and resurrected Christ.”

The London mayor, Sadiq Khan, said the 9/11 terror attacks “changed our world for ever”.

“Today, we remember the innocent people who lost their lives – including the 67 Britons, many of whom were Londoners.

“Our values of freedom, tolerance and respect will always, always prevail. Hate will never win.”

Relatives who lost loved ones in the attacks will attend a special memorial service at the September 11 Memorial Garden in Grosvenor Square in London on Saturday.

The names of those who died will be read at the private service, which has been organised by the September 11 UK Families Support Group, and a minute’s silence will be held.