European Commission President Jean-Claude Juncker is pushing for quick negotiations on the U.K.'s exit from the European Union, implying he doesn't want to wait until David Cameron steps down as Britain's prime minister this autumn.

He told journalists that the exit process should start "as soon as possible, however painful that process may be. Any delay would unnecessarily prolong uncertainty.''

He added: 'We stand ready to launch the negotiations swiftly, swiftly with the U.K.''

Earlier at 10 Downing Street in London, Cameron had said he would step down by October so that a new Conservative prime minister would lead the exit negotiations, thereby injecting more uncertainty into the process. He said it would then be up to his successor to invoke Article 50 -- the two-year process of negotiating a departure from the EU under the Lisbon Treaty, the bloc's governing legislation.

The two-year divorce process applies only to ending the current relationship with the EU. It does not apply to striking a new trade deal with the EU which would have to be negotiated separately.

Other major players sang from different hymn sheets, with German Chancellor Angela Merkel calling for "calm and reflection" and French President Francois Hollande urging that "the procedures foreseen by the treaties will be rapidly applied."

Earlier in Brussels, as Europe was still waking up to a post-Brexit hangover, the heads of the EU's other institutions called for sober calm and unity, as did the NATO secretary general.

"It is a historic moment, but for sure not a moment for hysterical reactions," said EU President Donald Tusk, speaking at the Justice Lipsius building where EU summits are held. "As you know, the EU is not only a fair-weather project."

"Today, on behalf of the 27 leaders, I can say that we are determined to keep our unity as 27," added Tusk, who served as Poland's prime minister from 2007 to 2014. "For all of us, the Union is the framework for our common future.

He also offered reassurances that there will be "no legal vacuum," saying that "until U.K. formally leaves the European Union, EU law will continue to apply to and within the U.K., and by this I mean rights and obligations."

Already next week, Tusk plans to propose that leaders of the EU's 27 other member countries begin a "wider reflection on the future of our Union," starting with an informal gathering in the margin's of next week's previously scheduled EU summit.

A bit later at the European Parliament down the road, the assembly's president Martin Schulz spoke of a "difficult moment" for both sides.

He added: "We have to now, from a legal and from a procedural point of view, assess what are the necessary next steps," including what the vote means for triggering Article 50.

Schulz said that EU legislators would hold an extraordinary plenary session next Tuesday, where they will vote on a resolution analyzing the outcome of the U.K. referendum and the way forward.

And at NATO on Friday, Secretary General Jens Stoltenberg said the U.K.'s position in the organization remains unchanged, and that the country will remain a "strong and committed ally, and will continue to play its leading role in our alliance."

He also said that NATO remains committed to closer cooperation with the EU, and will step up at that cooperation at next month's summit in Warsaw "because together we are more effective in upholding our common values and keeping our nations safe."

Like other political heavyweights who had spoken out in favor of the U.K. staying the EU, Stoltenberg also scrambled to do a bit of damage control, after telling The Guardian newspaper just before the referendum that a strong U.K. in a strong NATO is good for both sides and warned that a fragmented Europe will add to instability.

"Today, as we face more instability and uncertainty, Stoltenberg said in a statement Friday, "NATO is more important than ever as a platform for cooperation among European allies, and between Europe and North America. A strong, united and determined NATO remains an essential pillar of stability in a turbulent world, and a key contributor to international peace and security."

Image placeholder title

See full coverage of the Brexit debate here.