They were wrong. The UK voted to leave, kick-starting a period of political chaos and economic uncertainty. Conservative government ministers have repeated the phrase "Brexit means Brexit" to the point of parody - but what does Brexit actually mean?
Formal talks about the process of exiting have begun - but, on the first anniversary of that seismic referendum result, there is still little clarity about what the future holds for the UK, which until recently prided itself on its stability.
Here, four experts talk about how the UK has changed, and how it will continue to do so.
'Brexit is a symptom and a cause of the state of British politics'
Sophie Gaston, head of international projects at Demos think-tank
"Brexit is both a symptom and a cause of the state of British politics. It was a vote that laid bare profound divisions in ideologies, life experiences and social attitudes throughout British society in a powerful, cataclysmic way.
Its revelations will shape our politics for years to come - as politicians scramble to respond to the huge wave of anti-establishment feeling it has revealed. It has also created new opportunities for political realignment, as our society becomes increasingly polarised around age and education, and traditional political affiliations become less important.
There is a sense now that the recent election was a rejection of the 'hard Brexit' approach set out by the prime minister - certainly it cannot be seen as an endorsement of her programme, but Brexit is only part of the story.
The negotiations will involve significant compromises and tough decisions and, as talks progress, we can expect to see these playing out on the home stage.
In the longer term, the process of repealing EU legislation will put great swaths of policy issues on the table for debate in Britain for the first time in decades; without a clear mandate in parliament, it will be tremendously difficult for any prime minister to navigate this with confidence."
'Inequality will increase'
Faiza Shaheen, director of CLASS think-tank
"A lot of projections show that inequality is likely to increase over the next few years. The economic effect of Brexit is already starting to kick in because of the inflation impact of the pound. There are some signs of slow-down and falling confidence.
Inflation - increasing prices in food and so on - always hits the poorest most. That affects inequality. Uncertainty impacts investment from the private sector. There's all that in the short term. Over the long to medium term, it really depends how deals are negotiated.
The big question about jobs is which sectors are going to be protected. What happens to manufacturing will affect the job market for people without graduate degrees. What kind of trade deals are we making? Are we continuing current trends by going for high-end manufacturing and letting low-end manufacturing move abroad? We already have a lot of regional inequality and certain regions of the UK, such as the northwest, could be hit hard.
There are things we can do to mitigate the impact - investing in infrastructure, greening the economy, and having a big push on skills and education.
The thing is, if we are talking about reducing inequality, we're not just trying to protect what's there - we actually need to reverse current trends. We need a Brexit that is progressive, not even one that protects the status quo. But for all the talk of an ambitious Brexit and of jobs, there's no sense of a serious push on inequality. That's why it's very likely we'll see inequality increase - it simply doesn't seem to be a priority."
'It could have a detrimental effect on Britain's ethnic minorities'
Zubaida Haque, research associate at the Runnymede Trust
"We entered into Brexit talks because the Conservatives were interested in Britain's membership of the EU, but the reality is the political campaign was a proxy to talk about immigration. It became much more about racism and xenophobia than the universal issues that interested the British population. The negative and toxic language in the Brexit debate was really harmful.
The manifestation of that was apparent in the spike in hate crimes, which went up substantially after the Brexit campaign, about 20 percent in terms of racial and religious hate crime.
The situation isn't improving, and it wasn't great to begin with - hate crime statistics were already pretty poor, but it got much worse.
At the moment we're talking a lot about the rights of settled EU migrants being guaranteed, but what's missing from that debate is what about ethnic minorities in this country? What we've seen increasingly since Brexit, and perhaps even before, is universal issues around poverty, inequality and racial discrimination becoming submerged as Brexit has dominated the political agenda so much. We want a race equality audit of all the Brexit policies.
There's a fear at the moment that Brexit could have a detrimental effect on Britain's ethnic minorities, who have already been disproportionately affected by austerity policies. It's cumulative. Brexit is tied into questions about who we are, national identity. There's a lot of trepidation - and in surveys, over half of British people think that Britain has become less tolerant since the referendum."
'It could be a race to the bottom for the environment'
Rosie Rogers, political adviser at Greenpeace
"Brexit will have huge implications on environmental policy. Firstly, many of the best environmental regulations come from the EU, so what replaces them within UK law is now up for grabs.
The UK used to be known as 'the dirty man of Europe' until EU laws cleaned up our sewage and beaches. Now there is no guarantee that what the UK will come up with will be better than the EU standards.
Secondly, with Brexit on the top of the political agenda, there are worries that high environmental standards will be traded away at the negotiating table in favour of trade or immigration perks.
This means the UK might have a race to the bottom rather than the top for environmental regulations that, for example, keep our bees alive, our fish stocks healthy, and our beaches clean. Lastly, issues like air pollution, tackling global climate change and species like migratory birds don't respect borders and many of the key environmental issues we face need international cooperation.
The UK government must ensure to show global leadership on these issues so not to drag the EU or other countries down. With the ever-changing UK political scene it's not clear if this government has our environment at the top of the priority list, so it's up to us to make sure it is."