Prominent Brexit backers in the Tory party also expressed concern about Mr Cameron’s plan to let the situation drift until he formally leaves Downing Street in October.
Leading Leave campaigner and Tory MP Bernard Jenkin insisted it was unrealistic to wait until the autumn.
“I think there is a problem just saying ‘right, we’re not going to do anything between now and October’ to address the European Union relationship, because they don’t want us hanging around and destabilising all their arrangements. They want certainty, and I think we should be able to do this more quickly,” he said.
The call chimed with the views of European parliament president Martin Schulz, who warned the EU would not be held “hostage” while the Tory party squabbled over its next leader.
Mr Schultz insisted that uncertainty was “the opposite of what we need”, adding that it was difficult to accept “a whole continent is taken hostage because of an internal fight in the Tory party”.
In a blunt sign of Britain’s new status in the departure lounge of the EU, it emerged leaders of the remaining 27 member states would meet to discuss the emergency situation without Mr Cameron present on Wednesday.
The move came as embattled Labour leader Jeremy Corbyn was set to face down a growing wave of criticism against his role in the referendum with a speech on immigration and the consequences of Brexit.
Mr Corbyn has rejected calls for his resignation amid growing fury at his failure to galvanise the party’s traditional supporters to turn out for Remain as many Labour MPs express concern at the possibility of him leading the party into a snap autumn general election.
The Labour leader is facing a vote of no confidence from some of his MPs, with one senior source describing the mood within the party as one of “utter devastation”.
Former shadow minister Tristram Hunt told BBC Newsnight he was “not convinced” Mr Corbyn had the abilities to press the Labour cause in the looming negotiations with the EU on withdrawal.
After what some observers described as one the most significant days in British politics since the end of the Second World War, the Tory party was coming to terms with the ramifications of the pro-Brexit vote which forced Mr Cameron into announcing his slow-motion resignation, and saw SNP leader Nicola Sturgeon make it clear she was preparing to push for a second Scottish independence referendum.
Northern Ireland, which like Scotland, voted heavily to Remain in the EU, also saw calls from Sinn Fein for an all-island referendum with the Irish Republic on reunification.
With ex-London mayor Boris Johnson firm favourite to succeed Mr Cameron at the Tory helm, Labour’s Ken Livingstone attacked media “obsession” with the pair , which he said drowned out the real issues in the debate.
“It was like the whole of the media was obsessed by this sort of struggle between Cameron and Johnson as they gnawed away at each other’s testicles.
“It was like civil war in the Tory party, not a debate about the economy,” Mr Livingstone told Sky News.
The 52% to 48% victory for Brexit in the knife-edge referendum caught financial markets and the international order off-guard as repercussions from the move sent shock-waves across the globe.
As the financial impact of the vote continued to reverberate across the world, credit rating agency Moody’s moved to change the UK’s long term issuer and debt ratings to negative from stable due to the uncertainty unleashed by the result.
Mr Cameron, flanked by wife Samantha, made an at times emotional address on the steps of Downing Street as he announced his resignation.
The move saw calls from anti-EU Tories for a more Brexit-looking Cabinet to be put in place soon as questions were raised about the future of Chancellor George Osborne, who kept a very low profile in the wake of the result.
As the scramble to replace Mr Cameron gathered pace, Home Secretary Theresa May and Education Secretary Nicky Morgan were being talked of as potential rivals to Mr Johnson.
Meanwhile, ex-Labour PM Tony Blair’s former chief of staff Jonathan Powell floated the idea of a second referendum, telling BBC Newsnight: “It seems very unlikely at the moment, the EU saying no to it, the Brexit campaign saying no to it, but that is one option when we go forward and when people realise quite how ghastly the alternatives are.”
The Lisbon Treaty sets out a strict two-year timetable for withdrawal negotiations once a member state triggers article 50 and signals the intention to withdraw.
Tory former chancellor and Remain campaigner Ken Clarke told BBC Newsnight the country needed a balanced Government and a “balanced” leader.