Prime Minister Theresa May said her new government will fight "burning injustice" and "build a better Britain," in her first public remarks since succeeding David Cameron on Wednesday.

"If you're from an ordinary working class family, life is much less secure than many people in Westminster realize," May said, referring to the British parliament.

"The government I lead will be driven not by the interests of the privileged few, but by you," she said outside her prime minister's residence in London's 10 Downing Street.

"Following the [Brexit] referendum, we face a time of great national change," May said.

"As we leave the European Union we will forge a bold new, positive role for ourselves," she said. "Together we will build a better Britain."

Cameron resigned after he failed to convince a majority of voters in a June 23 referendum on EU membership to oppose leaving the political bloc.

May, 59, paid tribute to him as "a great, modern prime minister" who had stabilized the economy and reduced the budget deficit.

She stressed the full name of her party, the Conservative and Unionist Party, in an appeal for national unity.

"That word Unionist is very important to me," she said.

"It means we believe in the union, the precious, precious bond between England, Scotland, Wales and Northern Ireland," she said.

It also means uniting people by "fighting against the burning injustice that if you're born poor, you will die on average nine years earlier than others."

She vowed to tackle the disadvantages suffered by black people in the criminal justice system, poor white boys in education, women at work, people with mental or physical health problems, and younger people who "find it harder than ever before to own your own home."

In his last speech before formally tendering his resignation to Queen Elizabeth II, Cameron avoided mentioning Brexit and insisted he "leaves our country stronger." 

"It's not been an easy journey and, of course, we've not got every decision right, but I believe we leave our country stronger," he said.

"I believe Theresa will provide strong and stable leadership," Cameron said.

"It's been a privilege to serve the country that I love," he tweeted en route to Buckingham Palace.

In his final bow in parliament earlier Wednesday after six years as prime minister, Cameron said "I was the future once," echoing his 2005 comment to former Labour leader Tony Blair.

In a last prime minister's questions filled with jokes, laughter, tributes and farewells, Cameron told lawmakers from all parties that "you can achieve a lot of things in politics. You can get a lot of things done."

"Nothing is really impossible if you put your mind to it. After all, as I once said, I was the future once," he said. 

In 2005, in a remark aimed at Blair in his first prime minister'squestions as opposition leader, Cameron said: "He was the future once." 

May backed Cameron's Remain campaign before the referendum. But she faces mounting pressure to begin talks on Brexit with European Union officials sooner than she had planned.

She had advocated waiting until the end of this year to invoke Article 50 of the Lisbon Treaty, which sets the rules for a nation leaving the EU, but many British and EU politicians have urged her to move more quickly.

Much attention will now focus on what roles, if any, May hands prominent Brexit campaigners Andrea Leadsom, Boris Johnson, Michael Gove, Chris Grayling and Liam Fox in her new cabinet, amid speculation that one of those five could be given a new post of Brexit minister.

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