Egypt has a "precious moment of opportunity" to move towards "civilian and democratic rule", Prime Minister David Cameron has said.
His comments followed the resignation of Hosni Mubarak as Egypt's president after 18 days of protests.
Mr Cameron said the UK was a friend of Egypt and stood ready to "help in any way that we can".
Amnesty International is arranging a "day of solidarity" with the Egyptian people in central London on Saturday.
Also in London there were celebrations at the Egyptian embassy in Mayfair and at the heart of the Arab community around Edgware Road.
Speaking on the steps of Downing Street, Mr Cameron said Egypt's new government should start to put in place "the building blocks of a truly open, free and democratic society".
Mr Cameron said: "What has happened today should only be the first step.
"Those who now run Egypt have a duty to reflect the wishes of the Egyptian people and, in particular, there really must be a move to civilian and democratic rule as part of this important transition to an open, democratic and free Egypt."
Labour leader Ed Miliband said the protesters had "won a famous victory".
He offered his congratulations before adding: "Now the task must be to create that democratic future that people have today won."
Announcing Mr Mubarak's resignation, Vice-President Omar Suleiman said the president had handed power to the army.
Foreign Secretary William Hague called on Egyptians to settle their differences peacefully.
"The Higher Council of the Military Forces has a particular responsibility to implement the concrete and irrevocable steps this transition requires and to prepare for free and fair elections," he said.
The Amnesty event in Trafalgar Square had been intended as a show of support for the protesters in Egypt but following Mr Mubarak's departure organisers say there will now be "some celebratory aspects".
Speakers include human rights campaigner Bianca Jagger, and a big-screen television link is planned with Tahrir Square which was at the centre of the demonstrations in Cairo.
In London, Amr El-Bayoumi, 45, an Egyptian international lawyer, said he was "elated" to hear of Mr Mubarak's resignation.
He said: "My head is raised high because of my fellow Egyptians who have proved the power of the people in fighting for their rights and bringing down a brutal dictator who has been supported for decades by the US.
"We have seen a shining example for the world to see true democracy and true power to the people."
However, Mr El-Bayoumi, who recently returned from two weeks of protests in Cairo and planned to fly back to his native country on Saturday, added that Egyptians have "so much work to do to rebuild the country".
"Obviously, we need to see who will take power. We have a lot of work to do to get our society back from this ugly corruption and ugly brutality."
Ahmed El-Mokadem, founder and patron of the British Egyptian Society, said he was "delighted" that Mr Mubarak has resigned.
He said the Egyptian people had "lost confidence" during Mr Mubarak's tenure as president, which had halted the country's development.
Mr El-Mokadem said: "This is good not only for Egypt, the Middle East and democracy but also for the West. The West should not be worried at all because a developed Egypt that is not corrupt will only have one ally and that will be the West."
Mr El-Mokadem said his only fear was that the army could take control.
However, he added that this was only a "slight concern" because the army had been involved "from the bottom up" by helping protesters to force the president to step down.
Earlier, the UK Foreign Office has said it is "deeply concerned" about a Briton missing in Egypt.
Foreign Office Minister Alistair Burt said Hisham Morsi was last seen being removed from Tahrir Square on 31 January but it is unclear by whom and authorities have been asked whether they are holding him.
A Foreign Office spokesman said: "Hisham Morsi is the only British national that we are aware of who is missing at the moment."
Mr Morsi, who is of dual British and Egyptian nationality, is understood to be aged about 50 and based outside the UK, "probably in Cairo".