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Bottom line: Tourists, visitors, and transit cases are not affected by the UK's vote to leave the European Union.

The relevant issues are about asylum seekers and some types of EU nationals who seek to gain (or persist) settlement in the UK. If you are an EU national exercising derived rights or exercising treaty rights, please use Expats for your questions/answers.

Update 19 August 2016

The Home Affairs Committee published a report dated 27 July 2016, The work of the Immigration Directorates (Q1 2016), which concludes there is an "absence of certainty" over Brexit. It mainly deals with the status of EU nationals who are now exercising treaty rights in the UK.

Update 2 July 2016

What a lot of people are interested in now has a name: "domestic disentanglement from EU law". The House of Lords has added a briefing to their library...

Repealing and Reviewing Domestic Legislation—As part of the process of
leaving the EU, decisions would need to be made about how to deal with
existing domestic legislation passed to enable EU law to have effect
in the UK, a process which the House of Lords European Union Committee
has described as "domestic disentanglement from EU law". Parliament
would have an important role to play in reviewing, repealing, amending
and replacing legislation, a process which is predicted by many to be
complex and time-consuming. Once the UK had formally triggered Article
50, its timescales would apply independently of Parliament approving
domestic legislative changes associated with leaving the EU.

The full briefing is here.

Update 28 June 2016

The House of Commons has admitted a research paper to their library: "Leaving the EU: How might people currently exercising free movement rights be affected?"

On the 24th of June this paper was admitted: "Brexit: what happens next?"

Now that the UK has voted to leave the EU, what will happen next? This
Commons Library briefing paper looks at the immediate consequences of
the vote and some of the longer term implications. This paper
considers various questions about UK withdrawal from the EU and what
is likely to happen in the coming weeks and months. The issues include
the method of leaving the EU, continuing parliamentary scrutiny of EU
business and the withdrawal negotiations, and the implications of
Brexit for Scotland and Gibraltar.

Original Answer

Changes that will directly affect British nationals...

The British passport has "European Union" on the front cover. It will remain valid during the negotiations and a new design will most likely be phased in as existing passports expire;

The European Health Insurance Card (Ehic) (which enables British nationals to get access to medically necessary, state-provided healthcare during a temporary stay in any of the 28 EU countries, Iceland, Lichtenstein, Norway and Switzerland, under the same conditions and at the same cost (free in some countries) as people insured in that country), will remain valid during the negotiations and most likely be phased out as the UK negotiates separate treaties like those already in place with Australia and New Zealand;

The British driving permit will remain valid during the negotiations. Because it has an EU symbol on it, a new design will most likely be phased in as existing permits expire;

Spouses and long-term workers will most likely be grandfathered. There are no mass deportations of EU nationals envisioned.

See also: How will Brexit affect your finances?

Changes that will (or might affect) the family formation immigration routes...

A large part of the Brexit debate was focused on immigration. I will add a few things that are in the crosshairs. Indeed some of the rulings that help prompt the Brexit vote...

• 
  The Surinder Singh route. They never liked this ruling and it
  has acted as a thorn in the government's side for a long, long time.
  It's in the crosshairs and I think we can be sure that this
  immigration route will be closed, perhaps not this year, or even this
  decade, but it will be high on the list of priorities.


• 
  The Zambrano case. This is another case that never sat too well
  with the government. The government (two governments actually) were
  recalcitrant following the court's decision and it took a long time
  for them to even publish the guidance.


• 
  The Metock case. The UK's reaction was heated and recalcitrant but they finally implemented
  the ruling. Brexit advocates have seen this case as an extension of
  Singh (above) and hence a humiliating loss of sovereignty.

There are other rulings from the European Court that are similar and the UK has fought against them and opted out where they could.

These things affect boyfriends and girlfriends in long-distance relationships where they need an inward migration route because they cannot meet the rules. And overall these are about family formation and do not affect travellers and holiday makers to the UK.

Changes that affect tourists and visitors...

Remember that part of the mission of UK Visas and Immigration is to get lots of visitors in to the UK because the UK economy relies upon visitors, and this part of their mission will not change. If anything they will expand programmes like they are doing for China and India.

The UK has voted to leave the EU. How does this affect people
traveling to the UK and vice-versa?

The outlook for tourists visiting the UK looks great! Sterling is at an all-time low so holidays will be cheaper, and visitors will be especially welcome because it boosts the economy.

Are other visa-free nationals (such as US/Canada citizens) affected?

No change. The UK has always operated at arm's length from the Schengen system. In about 4 or 5 years you will see the "EU Nationals" queue at the airport become rebranded into something that does the same thing but without the EU logo. And the EU rules about what you can carry and the HMRC customs declaration exits will be rebranded. It's reasonable to expect the government to adopt the EU customs rules into UK law. But this will (most likely) be done by statutory instrument so nobody will notice the change.

What other consequences might 'Brexit' have on travelers?

The first change a traveller is likely to see will be in the duty-free shops. Unless renegotiated, customs limits are likely to be introduced and of course any EU branding will be removed.

Update 25 June 2016

Laura Devine (a boutique immigration firm catering to high-net-worth individuals and frequent adviser to Parliament) posted this update yesterday...

Notes and comments...

Note: maybe you like the Singh Route, maybe you don't. Maybe you agree with what the government will do, maybe you don't. The point being that it's in the cross-hairs and there are strong public expectations that something will be done.

Side note: It's also reasonable to expect that EU citizens who have been here for a long time will be grandfathered in one way or another. Removals/deportations of people who are working in the economy is not in scope at all.

Comments: Addressing a comment from Relaxed (to whom thanks) on the feasibility of reversing the Singh ruling...

@JonathanReez I don't think so, and that's actually one of the few
things we do know IMO. But that's a discussion we should have on the
chat, the point is that assuming this kind of piecemeal adjustments are
possible at all is highly speculative at this point. – Relaxed 11
hours ago

The practicalities and mechanics of actually reversing Singh are out-of-scope. The point being that the "immigration debate" surrounding Brexit was not about tourists and visitors.

Nothing for quite some time. No law has changed. This was merely a non-binding referendum.

Presumably the executive will agree with the people and Article 50 will be triggered by the PM, but even then, this is the first time it's ever happened so 1) it'll take a couple of years to sort out and 2) people aren't exactly sure how it'll happen.

As a result - the UK is still in the EU, no laws have changed yet, so for now the only way it might affect you is the exchange rate - the British pound has plummeted with the news.

As far as the EU treaties are concerned, the referendum does not cause a Brexit. To cause a Brexit, the PM has to officially inform the other EU members that the UK is leaving. This notification may be several weeks or months in the future. The two-year period mentioned later starts at that point.

With this notification, the official negotiations on the future relations between the UK and the remaining EU nations begin. They can end in different ways.

• If there is an agreement between all parties, they can sign it and decide when it will take effect. Presumably this would include new treaties on travel, commerce, etc.


• If there is no agreement after two years of negotiation and both sides agree to extend the negotiations, they may do so.


• If there is no agreement after two years and no agreement to continue negotiation, the Brexit will take effect. There would be no new treaties on travel, commerce, etc.

So in theory the Brexit could take effect the day after tomorrow. In practice this is highly unlikely. The brexit could take effect at any time in the future, if talks drag on but do not fail completely.

The Brexit agreement would define the new rules and their starting date. I cannot believe that either side would enact significant changes for tourists on short notice. Also, I consider it probable that there would be reciprocity in the new rules. Since the UK would not want to disrupt the Common Travel Area, visa-free travel would have to remain possible. The right to work would likely change, but that's for Expatriates.SE.

(All this can be found in answers and comments, but I thought I'd bring it together.)

TLDR

It is too early to tell if anything at all will be affected; since the referendum is just a notification of the will of the people. It is up to the government to act on it and initiate proceedings.

If the UK does proceed with the legal process of divorcing from the EU, it may be able to negotiate separate agreements that can have an impact on the freedom of movement.

Nothing will happen for at least the next two years; because that's how long it will take for the famous Article 50 to kick into effect.

Even then, individual agreements/concessions may be negotiated by the UK with the EU regarding the freedom of movement.

Now, suppose that UK is ejected from the EU, then I see the following impact:

1. For people that currently need a UK visa who are non-EU citizens, there will be no change in their situation.




2. For EU citizens, unless there is reciprocal agreement between UK and that particular EU member state, or UK and the EU in general regarding freedom of movement, you may need a visa to travel to the UK.




3. Any special concessions or waivers that were in place regarding importing of goods and any import fees - these may be affected if the UK is removed from the EU - as it would not be part of the single market. However, again, the UK may manage to negotiate to keep these agreements with the EU.