Video report by ITV News Correspondent Dan Rivers
The first lorries have been turned back as drivers face new enhanced paperwork checks to cross borders on the first day of the UK's new relationship with the EU.
The UK has formally left the EU's single market and customs union, with the country now set to pursue new trade deals as an independent nation.
ITV News Correspondent Dan Rivers reported Kent Police officers have been on patrol to enforce the new paperwork and issue fines for those missing the correct forms, which includes an access permit to the county.
The Bank Holiday has ensured Dover's road have been largely quiet on Friday, but concerns will remain over the coming days.
Six lorries trying to get to the Republic of Ireland were turned back on Friday morning in Holyhead, Wales, for not having the right papers - which may be a sign of what's to come elsewhere, Rivers reported.
ITV News Correspondent Dan Rivers reports from Dover
The first lorry leaving the UK for France was loaded onto the Eurotunnel without delay early on Friday morning.
And in Ireland on Friday morning, the first ferries arrived at Dublin Port at 5.55am, having travelled from Holyhead.
The UK's new relationship with Brussels was heralded with Big Ben sounding at 11pm UK time on Thursday - 12 midnight in Europe.
Four-and-a-half years since the 2016 referendum, the political squabble over Brexit cost two prime ministers their job before Boris Johnson delivered on his 2019 election pledge to "get Brexit done".
As seen in Dover and other Holyhead, the practical realities of the new relationship are already taking place, with customs checks in the UK and EU.
After 47 years as an EU member state, the UK has become a third country for trade purposes and customs declarations.
In his New Year's address, Mr Johnson said the EU had provided the UK with a “safe European home” during the 1970s, but the country has now “changed out of all recognition” with global perspectives.
His Christmas Eve deal with Brussels, which comes into effect immediately, allows for the continuation of tariff-free trade with the EU single market – though businesses and individuals will have to follow new rules.
The UK has reached several agreements with non-EU countries – such as Japan – to ensure continuity of trading arrangements for British companies from January 1.
Mr Johnson, writing in the Daily Telegraph, said the “great new deal” honoured the “most basic promises” of the 2016 referendum, and added that the UK has “taken back control of our money, our laws and our waters”.
“And yet it is also the essence of this treaty that it provides certainty for UK business and industry, because it means that we can continue to trade freely – with zero tariffs and zero quotas – with the EU.”
Under the new arrangements, freedom of movement rights will end, and while UK citizens will still be able to travel for work or pleasure, there will be different rules.
Passports must be valid for more than six months, visas or permits may be needed for long stays, pets will need a health certificate and drivers will need extra documents.
The automatic right to live and work in the EU also ceases, and the UK will no longer take part in the Erasmus student exchange programme.
Travel to Ireland will not change, but the Northern Ireland Protocol governing trade between Great Britain and the region entered into effect at 11pm.
It means Northern Ireland will remain in the EU single market for goods, and will apply EU customs rules at its ports, even though the region is still part of the UK customs territory.
The protocol will also see Northern Ireland follow certain EU rules on state aid and VAT on goods.
So what is and isn't in the deal?
Boris Johnson said the deal covers trade worth around £660 billion and that it is a "good deal for the whole of Europe", which means:
Goods and components can be sold without tariffs and quotas in the EU market.
It will allow the share of fish in British waters that the UK can catch to rise from around half now to two-thirds by the end of the five-and-a-half year transition.
Allegations of unfair competition will be judged by an independent third-party arbitration panel with the possibility of a “proportionate” response.
The Erasmus student exchange programme will be replaced in the UK by a worldwide scheme named after code breaker Alan Turing.
However, on financial services, a vitally important sector to the UK, Mr Johnson conceded he had not got all he wanted.
Gibraltar, whose sovereignty is disputed by Spain and Britain, will remain subject to the rules of the free-travel Schengen area, keeping the border with Spain open.
Government officials insisted the necessary border systems and infrastructure in the UK are in place, and they are ready for the “new start”.
The Prime Minister is understood to have marked the occasion with his family in Downing Street.
Eurosceptic MPs, once derided as fringe eccentrics on the margins of the Conservative Party, expressed their delight that the UK was finally leaving the customs union and single market.
Sir Bill Cash said it was a “victory for democracy and sovereignty” that could only be compared in peacetime with the evolution of modern democracy after the end of the Stuart dynasty.
Former Ukip leader Nigel Farage, who played a key role in the 2016 referendum, said December 31 is a “moment to celebrate 2021 as an independent United Kingdom”.
He said: “It’s a big moment in our national story and the end of a very long road for tens of thousands of us who fought against the establishment.
“We celebrated on 31 January when we left the European Union – tonight we leave the single market and the customs union.
“Yes, we spare a thought for Northern Ireland and our fishermen, but this is a moment to celebrate 2021 as an independent United Kingdom. It’s a shame the pubs aren’t open.”
The UK’s chief negotiation Lord Frost said the UK has a “great future before us” with the chance to “build a better country for us all”.
But Scottish First Minister Nicola Sturgeon, a vocal critic of Brexit, said: “Scotland will be back soon, Europe. Keep the light on.”
Mr Johnson’s Brexit trade deal was overwhelmingly backed by MPs and peers as it cleared Parliament on Wednesday before receiving royal assent.
Labour supported the deal, despite misgivings from some pro-European MPs, but all the other opposition parties opposed the agreement, including the Brexit-backing DUP.