U.K. Prime Minister David Cameron resigned after the result but said he would stay on until October when the Conservative Party conference is held. It is hoped a new leader can be elected then who will take charge of withdrawal process.
There is little appetite in Europe to delay the Brexit, however. European Commission President Jean-Claude Juncker said after the result that the process for Britain's withdrawal from the EU should begin "immediately" and French President Francois Hollande said there was "no going back" on the decision.
The German Chancellor Angela Merkel has called for calm, however, saying that the EU shouldn't draw "quick and simple" conclusions from the result. Donald Tusk, president of the European Council, also said it was not a moment for "hysterical reactions."
Merkel is due to meet with Hollande, Tusk and Italian Prime Minister Matteo Renzi in Berlin on Monday to discuss the next steps to take and David Cameron is travelling to Brussels on Tuesday where pressure is expected to be applied over the timing of a withdrawal.
Matthew Elliott, the chief executive of the Vote Leave campaign told CNBC that there was no need to rush the process. "I think quite rightly the prime minister has paused on that," he told CNBC on Monday.
For others, the content of the negotiations is key. Carsten Nickel, senior vice president at risk consultancy Teneo Intelligence, told CNBC on Monday that a solution would be arrived at if both the U.K. and EU compromised.
"I think that in the end it's going to be a typical European outcome, you have to find a solution and a middle ground and I think that is down to Angela Merkel … there needs to be a solution between the two ends of the debate – between the ones that want to move really quickly and the others who say 'let's take this easy and let's not rush to conclusions'."
Nickel said there were now "competing forces" at work in the EU with leaders split over the direction of the bloc – towards more political union (a founding EU principle) or more economic unity as a single market.
"There are really competing forces here, some will say that we need to consider this (EU) project as a single market and others say that we need to hold onto the idea of a political union. The key issue for me is that no one has really spent any time on thinking about how we move forward. They need to come up with answers now," he said.
He added that while the EU needed to strike a balance between maintaining economic ties with the U.K. and not jeopardizing the European project as a whole, the U.K. also needed to know exactly what it wanted from Europe.
"First of all, what the U.K. needs to do is make up its own mind about the single market - if you want to remain a member of the single market then you have to accept freedom of movement and that means immigration. That is the fundamental trade-off that no one has really thought about in the U.K. They need to make up their minds and that will inform how much they're able to get back in terms of economic integration from the European side."
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