May was visiting Scotland, England, Northern Ireland and Wales during her day-long tour, aiming to shore up support for the government's Brexit strategy.
Brexit remains a fractious topic, with former prime minister Tony Blair leading a push for second referendum.
May kicked off the trip with a visit to a textile factory in Ayrshire, southwest Scotland, before travelling to Newcastle in northeast England to meet a parent and toddler group.
"Brexit provides us with opportunities. I want to see us coming together, the four nations across the United Kingdom," she told the Ayr gathering, insisting that "we will be leaving the European Union on March 29 2019".
May was later to stop for lunch with farmers near Belfast in Northern Ireland before meeting Welsh business owners in Barry, then returning to London in time for tea with a Polish group.
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"I am determined that as we leave the EU, and in the years ahead, we will strengthen the bonds that unite us," May said before her visit.
"I have an absolute responsibility to protect the integrity of the United Kingdom as a whole.
Tensions with London
In a seismic referendum on June 23, 2016, 52 percent of voters in the UK opted for Britain to leave the European Union.
Most voters in England and Wales backed Brexit, while majorities in Scotland and Northern Ireland wanted the UK to stay in the EU.
There have been tensions between London and the devolved governments in Edinburgh, Cardiff and Belfast over May's handling of the Brexit negotiations so far.
Scotland and Wales last week backed bills to ensure that powers brought back from Brussels go to their capitals.
Another sticking point is the Irish border as Britain leaves the European single market and customs union.
All sides in the Brexit talks want to avoid imposing checks at the frontier with the Republic of Ireland.
May has agreed to Brussels' plan to keep Northern Ireland in the customs union if no better solution is found.
However, the proposition is deemed unacceptable by the province's pro-British and pro-Brexit Democratic Unionist Party, which props up May's Conservative minority government in the UK parliament.
'It's not too late'
Blair, who was Britain's premier from 1997 to 2007, believes Brexit to be a giant strategic mistake and is a leading advocate for giving voters an opt-out.
He argued that once the terms of its departure were known, voters should have the chance to choose whether they prefer that to EU membership.
"It's not too late until we leave," the former Labour leader told BBC radio.
"We keep this under debate until we actually see the terms of the new relationship and then we can decide whether those terms are better than what we have now."
A ComRes online survey of 2,019 British adults on Monday and Tuesday, published in the Daily Express newspaper, which is staunchly pro-Leave, found that 35 percent want a second referendum while 65 percent do not.
Polling expert John Curtice on Thursday told AFP that it was "still the case that a majority of people are opposed" to another vote, but that "the gap is narrowing".
Writing in the Express, Foreign Secretary Boris Johnson, who spearheaded the official Leave campaign, said that in a year's time, "Britain will be out of the European Union and re-engaging with the rest of the world.
"Like an unstoppable express, we are heading for Brexit, and frankly my friends, we can't arrive soon enough," he wrote.
"Our national journey out of the EU is almost over -- and a glorious view awaits."
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