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The Cameron government backed a "remain" vote, and from everything that can be known or inferred, expected a "remain" vote, and were unprepared for a "leave" vote.

The referendum was apparently intended as either a way to strengthen the legitimacy of the EU institutions in the UK by showing popular support for them, or to put pressure onto the EU institutions by showing a very important minority (but not a majority) of the UK population was intolerant of them.

Should the government have prepared for a "leave" vote despite believing it was not practically possible for "leave" to win? I believe so, absolutely, because without any belief that "leave" was possible, a referendum should not even have been initiated. However, hindsight is great, but the fact is that the government was, as the case often is, unprepared for what they weren't expecting to happen.

Concrete contingency plans with third countries would have been strictly worse

Part of the Brexit sales pitch was that the UK could have achieved better deals outside the EU. This turns out to be false, and indeed part of the point of being in the EU is to secure better deals.

Some attempts were made, but with no real success. India were not in a hurry. India wanted more migration access to go with any trade deal, and this was obviously a non-starter in today's UK.

No negotiation before notification

The EU simply refused to enter into such negotiation on grounds of principle.

No clear mandate

Prepare what, exactly? There have always been multiple conflicting "visions" of how Brexit should be done.

The whole thing was directly opposite to the way in which the Scottish Indyref was carried out, in which there was the SNP in power in the Scottish Parliament with a large clear manifesto on how it should be done. If that had been a success it was fairly clear what the plan was and who would be carrying it out.

The same was not true of Leave. Most of the key campaigners have never been in elected roles. UKIP have only ever had a couple of short-lived MPs. Moreover, there was no surge to UKIP in the 2017 snap election. Not only that, there were two "independent" Leave campaigns (although this may turn out to have been an attempt to cheat spending rules).

The Three Brexiteers

What a lot of people were expecting was that one of Gove, Davis, or Johnson would have taken over in the post-Cameron leadership election. They did not enter as candidates. This left May, a Remainer, to implement a plan she had no support for.

Tory Syriza

You cannot hold a referendum that obliges other countries to give you things.

Greece tried this at terrible cost and was ultimately unsuccessful. If there is a country with a case for leaving the EU, it is Greece, not the UK; but ultimately at the crunch time they decided they would have been worse off out.

One major issue was that, technically, the EU is responsible for all trade agreements of its member countries. They can not on their own negotiate such arrangements. The EU leadership has flat out denied any member country from meeting with the UK to even discuss such things. Since the EU has also tied immigration to trade, that pretty much drags many of the most important topics off the negotiating table. So even if there had been some magical way of predicting exactly that the UK was going to vote to leave, they wouldn't really have been able to do much in preparation beforehand anyway, atleast not in the way of signed agreements.

Quite apart from the illegality under EU law of an EU member negotiating a trade agreement with another country, it would simply have been impossible. The eventual long-term agreement between the UK and EU is certain to involve the retention of some EU rules and regulations, and no other trade deal can be negotiated until the UK knows exactly what those rules and regulations are. There’s no point, for example, in entering into complex negotiations with the US on relaxing food standards to allow easier imports of food from the US if there’s a good chance that the UK will then have to agree to keep following EU rules in order to be able to export food to the EU.

The idea when joining the EC back in the day was that it was a long-term or permanent arrangement. The Treaty of Rome did allow for further members to join but made no arrangements for member states to leave the ECSC. The idea for joining the EC was that a common market would generate economic properity etc. etc.—the same arguments that were made by Remain in the 2016 campaign. While there was always a certain anti-EC, later anti-EU sentiment in the UK there was never a majority for leaving in parliament (but they also didn’t vote on it as far as I know). Note that the idea of an ever closer union of the people of Europe was also already written into the Treaty of Rome in 1957.

It wasn’t until the Treaty of Lisbon that a withdrawal clause, the famous Article 50, was introduced into the EU framework. Prior to 2009, there was no reason to plan any type of withdrawal because there was no legal framework to do so. While territories had left the EC previously (Algeria upon independence and Greenland following a referendum) this did not result in member states withdrawing (because France and Denmark remained members); an important distinction.

This leaves us with the seven years between 2009 and 2016 to draft any provisional plans for what would happen after triggering the withdrawal clause of Article 50. But then again we have to remove the time until the 2010 election in which Gordon Brown was prime minister, him supporting the Remain side. So six years.

David Cameron was appointed as a prime minister following the formation of a coalition government in 2010 after that year’s election. However again, this was a person who favoured Remain and who had no intention of ever drafting a Leave agreement. Part of his re-election campaign included the promise of a referendum on EU membership in his second term. Following the 2015 election, his party gained a majority of seats in the House of Commons meaning that only now the question of a referendum was on the table—before that, it was a mere possibility in case the Tories won.

This leaves one final year between everybody knowing that there would be a referendum and the referendum actually happening. However, Cameron’s election promise was two-fold: he also believed that the UK would be able to negotiate a better deal to remain in the EU—he was able to negotiate such a deal and believed it to be strong enough to convince the electorate to vote Remain. Indeed, this deal was not fully negotiated until February 2016.

We were now mere months away from the actual referendum. A lot of political energy had flown into the idea of getting a better Remain. The prime minister was a supporter of Remain. While the leader of the opposition, Jeremy Corbyn, was criticised as not fully supporting Remain, the party as a whole did. Those people who campaigned for Leave either had done so for decades but were restricted to the backbenches or only switched sides in the runup. There was literally nobody around who could have drafted any reasonable leaving plan.

Of course, in June 2016 Leave won. The next morning, Cameron resigned; not only because he had no plans but also because this was not the result he had campaigned for and not the result he wanted to implement. A new government had to take over what some commentators perceived as shambles and start coming up with a plan. I shall leave it to the observer to decide whether they did well or not.

It is rather hard to make provisions for problems that are either not forseen at all or much harder than expected..

The real issue has been overlooked, maybe because from London the Irish border seems to be far away and the peace there has been holding for so long now.

It is therefore easy to miss the fact that the Good Friday Agreement i.e. in effect the open border was only possible because Ireland and UK were in a customs union.

Not once have I heard anyone mention, that in the end the peace was thanks to the EU.

Nor did I hear any suggestion how to keep the border open when the two sides are no longer in a customs union.

All other issues have been resolved but this one seems impossible; which is why they have agreed to postpone it for another two years. Will we see a practical solution by then? Doubtful, and from those doubts came the need for the 'backstop'..

Update March 2019

Brexiteers always have talked a lot about restoring the British sovereignty. I still do not notice in what I hear from the debates and comments in London that the need to control the borders (not just in terms of who comes across but also what goods and wares) is not just a necessity for the EU. It is just as much needed for UK's sovereignty, but no Brexiteer seems to have noticed that..

They say you should ‘hope for the best, but plan for the worst’, so why doesn't Britain seem to have done any such planning pre-2016?

But there were plans pre-2016 by both, the Leave and the Remain campaign. However, both were a bit unrealistic, partly because you don't know what you can negotiate before you actually do it and because neither side had an interest in being realistic and reasonable. Remember the £350 million a week pledge to fund the NHS by the Leave campaign, which was based on a simple deception based on confusing gross and net contributions to the EU budget? That was the level of analysis that was available pre-2016! One day after the referendum, the claim was basically given up.

That leaves the question if the UK government should have tried to negotiate before the referendum on behalf of both sides, to give the public a better understanding of the possible outcomes so that they could have made a more informed decision.

However, the EU could have simply refused to negotiate anything before Article 50 is even triggered. Or it could have negotiated in a very hard manner, not giving away anything, in order to convince the voting public in the UK that remain is the better option. It's the same argument why the UK government ruled out a second referendum after the first, in order not to weaken their negotiation position.

Article 50 by itself is not much, but it offers a plan. Two years to settle open bills and then trade on WTO rules. That is something. The settling of open bills or the negotiation of citizen rights went rather smoothly. The agreement that has been made between the EU and the UK on these issues will probably act as blueprint for others wishing to exit the EU. UK was a bit unfortunate to be the first one. Subsequent departures will have it easier there.

The real problems include the Northern Ireland question, which is still unsolved (there is only a backstop but not a solution) for all its complexity. I guess the UK government could have hardly solved it pre-2016, but surely could have highlighted the importance more, even if the public might have ignored it as an attempt of scaremongering. The other real problem is the future economic relationship regarding the often termed "cherry picking" approach. There nobody knows what is really possible and what is not possible. It depends on how dependent both sides are on each other and what they are willing to sacrifice (control, integrity) for gaining mutual benefit. That is not solved now, and hardly could have been solved pre-2016.

All in all, the UK navigates in unchartered water with the departure from the EU, it's simply the first to use Article 50. That's why the amount of preparations that you can do is limited. Other countries thinking about doing the same will have a much more realistic prospect of what it means and what is actually possible. For the UK, it was just bad luck to be the first one and they suffered from very biased and crooked referendum campaigns.

Other answers have answered aspects of the question. I want to add:

Article 50 was drafted with the intention that it not be used. The drafters aimed to make it so unfavourable to the invoking country, that no-one in their right minds would ever invoke it. This has been publicly stated by the chief official responsible for the drafting of the treaty.

This is why "the UK and the EU found themselves in a position where they had only couple of vague lines ... to go on."