A U.K. vote to leave the European Union on June 23 would change nothing and change everything.

On June 24, the country would still be an EU member, even if the government was in turmoil. While the ruling Conservative Party sorted out who would be its future leader—and therefore the country’s prime minister—legally everything would stay the same.

But after the dust settled, the government would face a critical question: How quickly should it pursue a formal exit from the bloc?

If it went for a quick departure, it would be seen as swiftly fulfilling the political orders of the British people; delaying could be highly unpopular among voters.

But it would face an economic conundrum that the U.K. Treasury pointed out last month: “a trade-off between securing a deal as quickly as possible to reduce uncertainty in the short term, and securing the best possible deal for the U.K. to minimize the economic costs of exit over the long term.”

Leave soon, and there wouldn’t be time to negotiate a favorable deal on post-exit relations; take the time to negotiate a better deal, and it would extend the period of uncertainty for consumers, investors, businesses and banks, increasing the likely hit to the economy.

After a leave vote, the U.K. would immediately enter a limbo land, on its way out of the bloc but still legally a member. It would still have a vote but its positions on broader EU policy would have no standing. British representatives in the corridors of power in Brussels would be like ghosts at the feast.

The views of British members of the European Parliament, its diplomats and politicians on future EU policy would be largely irrelevant. Questions would be raised about whether Jonathan Hill, Britain’s commissioner responsible for financial services, should retain such an important post.

The U.K. is scheduled to hold the rotating presidency of the EU in the second half of next year, a role in which it would be guiding large parts of the bloc’s legislative agenda.It is almost inconceivable that would happen.

For others in Brussels, there would be a strong incentive to bundle the U.K. out the door as fast as possible. The 27 other EU governments would all need to agree on the exit deal, as would a majority of the members of the European Parliament. Many would want to focus on addressing the bloc’s other big problems—but also to send a message to other would-be leavers that exit isn’t an attractive option.

“Leave means leave,” said one prominent MEP this week, saying his view reflected that of many senior politicians in his party grouping.

A British exit would be “a disaster for both sides” but it would be worse for the U.K. “We will show that leave is a mistake,” he said.

Because there would be no “special treatment treaty,” this person said, negotiating a British exit would much easier than the complicated process of joining the EU, that can take the better part of a decade or more. Cutting the relationship “is absolutely easy,” he said.

One decision is absolutely in London’s hands: when to begin the process of withdrawal by invoking Article 50 of the Lisbon Treaty. That envisions two years of negotiations over exit arrangements, which can be extended only by agreement of all 28 governments.

Prime Minister David Cameron said he would invoke the article immediately. But Mr. Cameron is unlikely to be prime minister for long if the vote goes against his recommendation to stay. And some supporters of leave argue the government should bide its time before triggering Article 50, at least to figure out what it wants to achieve in the negotiations.

The exit agreement wouldn’t be the only issue on the agenda. Preparations would have to start over the U.K.’s new trade relationship with the bloc and over what London wants in new trade ties with the rest of the world.

Some Leave proponents argue no such trade deals are necessary and that the U.K. should just cut import tariffs to zero, but most analysts believe this course of action is unlikely because it would subject the U.K. economy to enormous disruption.

Domestically, an assessment would have to begin overhauling the U.K.’s policies and domestic legislation in areas now under the EU’s purview.

The U.K. could quite conceivably leave the EU without having a trade deal with the bloc, which could come later or not at all, and it is unlikely to have even embarked in earnest on trade agreements with non-EU countries. These processes are going to take years, probably more than a decade, to conclude.

But an exit could well take place faster than many people now believe is likely. Caution suggests the U.K. should take time before it formally steps out of the bloc. However, other factors—British politics, the wish to reduce the period of economic uncertainty and the desire of the rest of the EU to move on—suggest that fateful date may be sooner rather than later.