Mrs May made a dramatic start as prime minister by "giving the boot to Chancellor George Osborne" - and appointing Brexit campaigner Boris Johnson as her foreign secretary, writes The Scottish Sun.
Nevertheless, the normally soft-spoken former French prime minister Ayrault had a strong warning for the new foreign policy chief of his near neighbour.
While Johnson will be the foreign secretary, he will not be the frontrunner in Britain's European Union talks.
David Cameron's speech before parliament Wednesday - his last as Britain's prime minister - was packed with moments both poignant and amusing.
Philip Hammond leaves Downing Street after being appointed Chancellor of the Exchequer.
But she "explained that we would need some time to prepare for these negotiations". This task has been allocated to the secretary of state, a position that's been filled by David Davis.
May spoke outside 10 Downing Street shortly after she emerged from Buckingham Palace, having paid the requisite visit to the Queen to be invited to form a government.
Mr Osborne had been fired because his "brand" was seen as "too tarnished", BBC political editor Laura Kuenssberg said.
Some ministers are remaining in post as May shakes up the Cabinet, including Defense Secretary Michael Fallon.
"You can get a lot of things done", he said.
Boris Johnson succeeded Philip Hammond in the Foreign Office. He's the embodiment of the phrase, "a safe pair of hands", and takes on the biggest role as Mrs May's supporter.
The French Foreign Minister described Johnson's appointment as a "sign of political crisis" in the United Kingdom following Brexit.
Britain's credibility is "hanging by a thread" after Mr Johnson's appointment.
"(He has) his back against the wall to defend his country but also with his back against the wall the relationship with Europe should be clear", Ayrault said.
Lib Dem leader Tim Farron predicted Mr Johnson would "spend more time apologising to nations he's offended" than working as foreign secretary.
Despite being a part of a Tory government that has disadvantaged working class people, May said her government "will be driven not be the interests of the privileged few, but by yours".
On her first full day in office, May dismantled Cameron's affluent metropolitan clique, dubbed the "Notting Hill set" after the former prime minister's trendy West London neighborhood.
She highlighted the "precious bond" between England, Scotland, Wales and Northern Ireland and between "every one of us".
Tributes to Mr Cameron, who was Prime Minister for six years and Conservative leader for nearly 11 years, poured in from party members in Oxfordshire.