Germany and France have led demands for Britain to negotiate a quick divorce from the European Union.
Paris is warning that populism will otherwise take hold after the vote to leave the bloc sent shockwaves around the world.
The European Central Bank added to the pressure by saying Britain's financial industry, which employs 2.2 million people, would lose the right to serve clients in the EU unless the country signed up to its single market -- anathema to "leave" campaigners who are set to lead the next government in London.
Britain's decision to leave the EU, the world's largest trading bloc, is the biggest blow since World War Two to the European project of forging greater unity.
In the country itself, divisions widened after the relatively close result in Thursday's referendum.
Scotland's first minister said she wanted to open negotiations directly with Brussels, saying the option of a second independence referendum should be "on the table".
South of the border in England, other pro-EU Britons also pondered their uncertain future, with the Daily Mirror newspaper asking: "So what the hell happens now?"
Prime Minister David Cameron announced on Friday he would resign after Britons voted 52-48 percent to exit the EU, a result which sent global stock markets plunging, and inflicted the biggest one day drop on sterling in history.
But Cameron promised to stay on as a caretaker until October while the Conservative Party chooses a new leader, saying it was up to his successor to notify the EU formally that Britain wanted to leave under the Lisbon Treaty, which provides for two years of divorce proceedings.
European leaders made clear they saw the possibility of months of uncertainty before serious talks begin as unacceptable for the 27 other member states.
"This process should get underway as soon as possible so that we are not left in limbo but rather can concentrate on the future of Europe," German Foreign Minister Frank-Walter Steinmeier said after hosting a meeting of his colleagues from the six founding members of the EU - Germany, France, Italy, the Netherlands, Belgium and Luxembourg.
French Foreign Minister Jean-Marc Ayrault warned of the dangers of delay.
"We have to give a new sense to Europe, otherwise populism will fill the gap," he said.
Eurosceptics in a number of member states have welcomed the vote by Britain to become the first sovereign state to leave the EU.
This has deepened fears among mainstream politicians about the rise of anti-establishment parties, not least in France where far-right National Front leader Marine Le Pen is expected to perform strongly in a presidential election next year.
This, they worry, could lead to a domino effect of other countries wanting to leave too.
Slovakia's far right People's Party on Saturday launched a petition for a referendum on the country's future in the EU.
European Commission President Jean-Claude Juncker also wants to begin negotiating Britain's departure immediately.
"It doesn't make any sense to wait until October to try to negotiate the terms of their departure," he said on Friday.
The British pound fell as much as 10 percent against the US dollar on Friday to levels last seen in 1985 on fears the decision could hit investment in the world's fifth-largest economy, threaten London's role as a global financial capital, and usher in months of political uncertainty.
The euro slid 2.0 percent against the dollar and world stock markets saw more than US$2 trillion (NZ$2.80 trillion) wiped off their value.
Ratings agency Moody's downgraded its outlook for Britain, saying its creditworthiness was now at greater risk as the country would face substantial challenges to successfully negotiating its exit from the bloc.
But the period of uncertainty risks damaging economies far beyond Britain's shores and European Central Bank policymaker Francois Villeroy de Galhau joined the clamour for haste.
"What happened on Thursday is bad news, first of all for Britain," he told France Inter radio, although he acknowledged there would be more limited "negative consequences for the European economy".
Villeroy delivered a warning to Britain over the City of London financial centre which handles trillions of euros of business through its traders and clearing houses even though it lies outside the ECB's jurisdiction.
That was at risk, including the "passport" arrangement under Europe's single market rules which allow London banks to do business with clients in the euro zone, even though Britain never joined the common currency.
The only way around this was for Britain to follow the example of Norway, which lies outside the EU but has joined the single market.
This means signing up to the rules, including the free movement of workers - likely to be taboo in Britain where the "leave" camp promised to control immigration from the EU, including huge numbers from the poorer eastern member states such as Poland.
Cameron has said someone else should take the lead in Britain's complicated exit negotiations.
His replacement could be his Conservative rival Boris Johnson, the former London mayor who became the most recognisable face of the Leave camp and who is now favourite to succeed him.
There was euphoria among Britain's eurosceptic newspapers.
"Birth of a new Britain," the Daily Telegraph said, while the Daily Star tabloid borrowed from Donald Trump's campaigning message with its headline "Now Let's Make Britain Great Again".
But those among the more than 16 million who voted to stay in the EU also raised their voices.
The United Kingdom itself could also now break apart.
Scotland's pro-EU First Minister Nicola Sturgeon said her devolved government was preparing to present legislation allowing a second independence referendum while continuing discussions on its place within the EU.
Scots rejected independence in 2014 by 55-45 percent.
But Scotland voted 62 to 38 percent to stay in the EU on Thursday, raising the possibility that it would be forced out of the EU against its will.
Elsewhere, more than 1.1 Britons have signed a petition on parliament's website calling for a second EU referendum because fewer than 60 percent of voters had backed Leave on a turnout of under 75 percent.
The petition will have to be discussed by MPs but has no legal force.