German Chancellor Angela Merkel has said it is "important that we still view Britain as a friend and partner", despite its decision to leave the EU.
But speaking as she arrived at a summit of EU leaders in Brussels, she insisted "there will be no informal or formal talks before Britain has used Article 50" of the EU treaty, which will formally start the process of negotiating its withdrawal from the 28-nation bloc.
She earlier said that although Britain will want to maintain ''close relations'' with the EU once it leaves, it cannot expect a business-as-usual approach.
"Whoever wants to leave this family cannot expect to have no more obligations but to keep privileges," she said, and added: ''We will ensure that the negotiations are not carried out with the principle of cherry picking."
French President Francois Hollande has urged Britain to start exit talks "as fast as possible" as he warned the rest of the union has "no time to lose" to plan its future without the British.
He acknowledged it is a historic moment as no nation has left the EU before, but he insisted "history continues" and said "Europe doesn't stop" with Brexit.
At the EU Parliament, EU Commission chief Jean-Claude Juncker said he is banning any informal and secretive negotiations on Brexit before the UK invokes Article 50.
He told a special session that ''I want the UK to clarify its position, not today, not tomorrow at 9am, but soon. We cannot allow ourselves to remain in a prolonged period of uncertainty''.
European Council chief Donald Tusk echoed that sentiment, saying the bloc will not be able to begin negotiations on Brexit until Article 50 is invoked, declaring that ''is the only legal way''.
He also said the EU will hold a meeting in the Slovak capital Bratislava in September to assess the EU's future in the wake of Britain's vote to leave.
Denmark's PM Lars Lokke Rasmussen, meanwhile, warned EU scepticism is not just a British issue, and said he hopes David Cameron "will be met with understanding" by his counterparts in the bloc.
Arriving at the EU summit, he said: "What we have seen in Britain, it could have happened elsewhere. We need to take this seriously."
But while he added that "it makes sense to admit that Britain needs time" to negotiate its exit, h is Swedish counterpart Stefan Lofven said "it would be great if the United Kingdom told us as quickly as possible what they want".
Greek prime minister Alexis Tsipras used the result of the vote to hit out at the EU, saying the British decision to leave was predictable because the EU has disregarded core principles of democracy for too long.
Mr Tsipras has been at loggerheads with the EU leadership for well over a year as he has negotiated stringent austerity conditions to obtain bailout funds for his country. During those times he often decried the lack of openness of the EU.
In Brussels on Tuesday, he said: "Europe has reached a predictable crisis because of the democratic deficit, because of the absence of social cohesion and solidarity."
He said he hopes "that the outcome of the British referendum will work as a wake-up call for Europe".
In a direct attack on EU policies, he said there is a need "to replace austerity with growth, to replace division with convergence, to replace unemployment with decent jobs, and endless negotiations behind closed doors with transparency and democracy".