The EU laws don't allow any official procedure to expel a member state from the union (they only make it possible to strip a nation of some voting rights etc.). The EU leaders are currently very impatient because a prolonged uncertainty about the status of the U.K. will energize "exit" movements in other countries. (Angela Merkel differs from the EU Commission etc., she says that there's no reason to hurry.)
But they can only use "unofficial" methods to persuade, pressure, threaten the U.K. by ad hoc sanctions etc. to act quickly, not any "standard" procedure. From the viewpoint of the EU law, the result of the U.K. referendum means absolutely nothing. As far as the EU laws are concerned, the British government could very well ignore the result of the referendum, too.
In fact, even within the U.K., the referendum is officially just an advisory one and its result will only become binding once it's confirmed by the U.K. Parliament. And there exist various efforts to organize new referendums about the return of the U.K. to the EU, separation of London and Scotland, and several other things, so things could be a bit ambiguous for a while.
At the same moment, politically, everyone who matters agrees that the result of the referendum is rather clear and cannot be ignored.
The legally expected steps now are the following: At one moment, the British government must send a letter to the European Council or make a speech saying that they formally activate the "Article 50" of the EU treaty (which is about the exit). From that moment, a 2-year window will be dedicated to negotiations. Some majorities of EU countries have to agree with the proposals, and so on. If the agreement isn't completed within the 2-year deadline, the treaties with the U.K. will cease to apply regardless of consequences at that moment of the year 2018.