Britain should probably formally quit the European Union around December 2018, new Brexit Secretary David Davis has signalled. He will help May in deciding when to start the two-year process of withdrawal, and how to reconcile her desire to retain access to the EU's single market with public pressure to curb immigration.
In a series of congratulatory phone calls after taking over as Prime Minister, Mrs May spoke to German chancellor Angela Merkel, French president Francois Hollande and Irish Taoiseach Enda Kenny.
May stressed her commitment to delivering Brexit but "explained that we would need some time to prepare for these negotiations and spoke of her hope that these could be conducted in a constructive and positive spirit", a Downing Street spokeswoman said.
May also appointed other Brexit supporters to major Cabinet posts, including former Foreign Secretary Philip Hammond as finance minister and Amber Rudd to May's old job of home secretary. He replaces George Osborne, whose determination to balance Britain's books made him synonymous with austerity.
The biggest surprise is the appointment of Boris Johnson, the Tory members' darling, as the foreign secretary - one of the greatest offices of state, with a hugely different role as the United Kingdom contemplates life outside the EU. Other cabinet members Oliver Letwin, John Whittingdale, and Nicky Morgan also left.
Davis, a former minister of state for Europe in the 1990s, will now be "Secretary of State for Exiting the European Union". Fox, another prominent Euroskeptic, will be secretary of state for global trade.
Mrs May has already made telephone calls to key worldwide partners, including President Hollande, and the Elysée says the French president told her he hopes negotiations for the UKs EU exit will now begin as fast as possible. "Brexit means Brexit" has quickly become her new mantra. The two had clashed over policing in London while Johnson was serving as mayor. The UK's Daily Telegraph predicted the Brexit under the new PM would be "tougher and more uncompromising" than it would have been under Mr Johnson.
Secretary of State John Kerry called Johnson to congratulate him and "stressed US support for a sensible and measured approach to the Brexit process", the State Department said.
Cameron stepped down after Britons rejected his entreaties and voted in a June 23 referendum to quit the European Union, weakening the 28-nation bloc and creating huge economic uncertainty, including likely damage to trade and investment.
"I suspect May will take a firm grip of her ministers and will lead the strategy", he said. An official photograph showed her curtseying to the smiling monarch.
Only on Monday did Mrs May learn that she would become Prime Minister, when the last remaining contender to lead the governing Conservative Party, Mrs Andrea Leadsom, quit the race.
She suffers from Type 1 Diabetes.
With Mrs May stating she would not be rushed into triggering Article 50, Mr Hammond said the Government would "consider carefully how we go about getting the very best deal possible".
She further reiterated that her party values unionism between all of the citizens and not just among England, Scotland, Wales, and Northern Ireland.
Outside Downing Street, a group of demonstrators chanted: 'What do we want?
Britons chose Brexit despite a barrage of warnings that severing ties would create huge uncertainty and plunge the economy into recession.
May is something of an unknown quantity internationally, but European Council president Donald Tusk said he looked forward to a "fruitful working relationship".
In a carefully orchestrated political ballet, Cameron made his final appearance as prime minister in Parliament, turning the usually raucous prime minister's questions session into a time for praise, thanks, gentle ribbing, cheers - and a sprinkle of criticism.