Her last rival, Andrea Leadsom, dropped out of the race on Monday, removing the need for a nine-week contest to decide who would become leader of the ruling Conservative Party and prime minister.
German Chancellor Angela Merkel said on Monday that Britain should clarify quickly how it wants to shape its future relationship with the European Union, adding she wanted London to remain an important partner.
The government must now get on with its job and inspire confidence.
May took the remark as a badge of honour, repeating it to lawmakers at a party leadership hustings, and adding that the next person who would find out just how hard she was would be European Commission head Jean-Claude Juncker.
Johnson said on his appointment: "We have a massive opportunity in this country to make a great success of our new relationship with Europe and with the world".
"Theresa May's virtual "coronation" as prime minister has delivered a boost to the pound as the clouds of uncertainty following the Brexit vote start to disperse", said market analyst Neil Wilson at ETX Capital, a financial trading company in London. May opposed the so-called Brexit referendum, but has said she will carry out the people's will.
Corbyn has clung to his job, citing support from the party grassroots members, and the 116-year-old party is now locked in a bitter power tussle that risks destroying it.
She promised to prioritize more house-building, a crackdown on tax evasion by individuals and companies, lower energy costs and a narrowing of the "unhealthy" gap between the pay of corporate bosses and their employees.
Before the referendum, leading scientists and academics had warned that leaving the European Union would cause a major funding blow to British universities and scientific research.
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The Conservative leader quit after Britain voted to leave the European Union against his wishes.
"It was very matter of fact, businesslike", said Kirkhope, also a former junior minister in Britain's interior ministry. "If she were a poker player I would probably have some reservations about taking her on".
May is a keen cricket fan and lists her hobbies as walking and cooking, telling one interviewer that she had more than 100 cook books at home.
Instead, a source who has worked closely with her told AFP on condition of anonymity she was "incredibly hard-working".
In an interview for BBC radio, she selected the Abba song "Dancing Queen" as one of her favourites.
"The Brexit vote plunges all companies - big and small, domestic and global - into the abyss of the unknown when it comes to their tax matters and how to plan for the future, but the key is not to panic", said Kevin Conway, a London-based tax partner at King & Spalding law firm.
Despite criticism for a poster campaign telling illegal immigrants to go home, May has largely won plaudits from both colleagues and political opponents during her six years as interior minister.
Damian Green, a Conservative lawmaker who worked under her as a junior minister, described her as "completely Stakhanovite" - a Russian term for someone with legendary capacity for hard work. "She wasn't to be pushed aside or pushed about".