Britain must launch the process of leaving the EU immediately and allow the rest of the bloc to forge ahead with greater union, EU leaders said on Friday.
British Prime Minister David Cameron, who announced his resignation on Thursday after the referendum result, said Britain would not make any formal notification before October, once his Conservative Party has chosen his successor.
However, President of the European Parliament Martin Schulz told The Guardian "I doubt it is only in the hands of the government of the United Kingdom."
"We have to take note of this unilateral declaration that they want to wait until October, but that must not be the last word."
In a joint statement, Schulz – along with President of the European Council Donald Tusk and President of the European Comission Jean-Claude Juncker – said postponing the UK's exit would "unnecessarily prolong uncertainty."
Manfred Weber, head of the largest political group in the assembly, the centre-right European People's Party, called on Cameron to trigger Britain's exit at an EU summit on Tuesday and to start leave negotiations immediately.
"We have the will of the British people on the table," Weber told reporters, saying it now had to be implemented "and the most important thing is that we do this very quickly."
Any EU country that wants to leave the bloc has two years to negotiate the terms of the divorce, starting from the moment it formally notifies the EU of its intention to exit.
But some Brexit supporters have suggested London could delay that notification to make time for informal talks on the best possible exit deal.
Weber said the EU could not wait for British politicians to squabble over who would be the next prime minister.
"We need to avoid a long period of uncertainty and the European continent cannot be occupied by an internal Tory [Conservative] battle about who will be the next leader of the Tory party and the prime minister of Great Britain," he said.
Guy Verhofstadt, a prominent European federalist who leads the third biggest bloc in the European Parliament, said he had no patience for "an internal cat-fight of the Tory party" and said the departure of eurosceptic Britain made it easier for the rest of the EU to unite.
"This ends a discussion of nearly 40 years with opt-ins, opt-outs, rebates, earmarks, you name it," Verhofstadt, leader of the centrist liberals, said, referring to special conditions that Britain has secured during its EU membership.
Brexit was a chance to transform the EU into "what the founding fathers initially had in mind," he said, "not a loose confederation blocked by a unanimity rule that always delivered too little, too late."
Faced with the loss of such a large member of the bloc and fears that the vote may inspire other eurosceptics on the continent, EU leaders may push for a quick show of unity, including on defence and security cooperation.