But the visit, which is also the one-year anniversary of her signing the Article 50 letter to leave the bloc, comes at a time when the country appears no less divided than at the referendum.
There are just months left to strike a deal on the future UK-EU relationship.
The Irish border remains a key difficulty in Brexit with no solutions yet as to how it might be addressed.
Since formal negotiations began between the two sides last June, an agreement has been struck on a Brexit "divorce bill" - but the crucial issue of how they will trade together has yet to be settled.
While today is a symbolic milestone, Mrs May will be looking ahead to the key date of October - by which time the European Union has said it wants to see a final Brexit deal completed.
Shadow chancellor John McDonnell defended Ms Thornberry's comments, emphasising her "sarcasm" in response to a journalist's question.
He told BBC radio: "We understand that we need to get the British public's support and that an extension would not be popular".
Media captionWith a year until our departure, what do voters make of the Brexit progress so far?
Starting her tour in Scotland, May met workers in a textile factory, focusing on the future benefits she said Brexit could bring for trade.
The Scottish and Welsh governments have both repeatedly claimed the UK Government's Brexit proposals are a Westminster "power grab", with powers returning from Brussels going to London instead of Edinburgh or Cardiff.
Mrs May insisted that each of the devolved administrations will see "an increase in their decision-making powers" as a result of the return of responsibilities now exercised by the EU. The prime minister will end the tour by meeting businesses in Barry, south Wales.
The EU maintains that by leaving its single market and customs union, Britain will be making trade more hard.
And Theresa May's message is suitably upbeat.
The UK construction sector is more reliant than most on migrant workers from Europe and at present, 9% of our construction workers are from the European Union, ' said Brian Berry, FMB chief executive, adding that in London, this rises for almost one third.
British prime minister Theresa May toured all four countries within the United Kingdom on Thursday (29 March), a year to the day before it leaves the European Union, vowing to "deliver a Brexit that unites" the country.
Even so, ministers are still working overtime to convince the devolved administrations they won't lose out after Brexit.
During her visit to Bangor, Northern Ireland, on Thursday, she repeated her "commitment to avoid a hard border" between the United Kingdom region and the Republic of Ireland, and to "protect" the 1998 peace agreement.
"The vote for Brexit had different elements to it - some people were voting due to their concerns about immigration, people wanted to take back control of our borders, our money, our laws", she said.