NEW PM Theresa May will make appointing a cabinet minister to take charge of Brexit one of her first tasks as she takes the keys to No 10.

Britain's new prime minister will swiftly begin drawing up her new top team and is expected to increase the number of women in government.

David Cameron will take to the Despatch Box in the House of Commons for his last session of questions as PM in the House of Commons before the historic shift of power takes place.

The outgoing PM - who will hand over the keys to Number 10 to Theresa May within hours - said nothing to reporters outside, but is expected to make a statement later setting out what he sees as his legacy before going to Buckingham Palace to offer his resignation to the Queen.

Mrs May will follow him to the Palace to be formally appointed his successor by "kissing hands" with the head of state, and is expected to make her first speech as PM outside the famous black door in Downing Street outlining her priorities for the new administration, returning to No 10 as the country's second female prime minister.

Mr Cameron told The Daily Telegraph it had been "a privilege to serve the country I love".

"I came into Downing Street to confront our problems as a country and lead people through difficult decisions so we could reach better times," he said.

"As I leave, I hope people will see a stronger country, a thriving economy, and more chances to get on in life. It has been a privilege to serve the country I love."

Removal vans were seen on Tuesday as Mr Cameron, wife Samantha and children Nancy, Arthur and Florence, packed up at their home since 2010 and prepared to return to a life outside the gates to Downing Street.

Taking his last Cabinet meeting, the outgoing PM spoke of his "honour and pleasure" at having served for six years and told colleagues Mrs May was "the right person to lead the country wisely through the difficult times ahead".

Although a Remain supporter, Mrs May has repeatedly stressed that "Brexit means Brexit" and the hunt for a building to house the department that will steer Britain out of the European Union is already under way.

After presenting herself as the unity candidate, the incoming leader is expected to offer plum posts to leading figures from both camps in the EU referendum.

Mr Osborne looks unlikely to stay on as Chancellor after the PM-to-be trashed parts of his economic legacy in a campaign speech on Monday, with Philip Hammond, who has long coveted the role at the Treasury, among those who could take over.

Senior Brexiteer Chris Grayling will be rewarded for his role running Mrs May's campaign.

Andrea Leadsom, whose shock withdrawal from the leadership race meant the expected nine-week leadership campaign was truncated to just a couple of days, is expected to be offered a job in recognition of her raised profile.

Big question marks are hanging over the future of Brexit standard-bearers Boris Johnson and Michael Gove, who were seen to have blotted their copy-books in the wake of the referendum result.

Mrs May, a founder of the Women2Win group to increase the number of female MPs, is keen to see more women in "prominent" roles.

A spokeswoman for Mrs May said: "Civil servants have already been charged with finding a building to house the Brexit department - an indication of Theresa's commitment to get on with delivering the verdict of EU referendum. Brexit means Brexit and we're going to make a success of it."

They added: "It was Theresa that set up the campaign to elect more female MPs to parliament - and she has always believed that there should be more women in prominent government positions."

Mr Grayling declined to say whether he was expecting Mrs May to reward him with a senior Cabinet post, telling the BBC Radio 4 Today programme: "I will do whatever - if anything - she asks me to do.

"I think that should be the same for everyone. This is about making sure we have got the right team for the challenges we face."

The leading Leave campaigner said it was right that the minister given responsibility for heading Brexit negotiations should be "somebody who was at the heart of the Leave campaign".

He said he was "absolutely confident" that Mrs May would make good on her promise to implement the result of last month's EU referendum.

"She will do this and she will do it well, but she will also do it carefully," he said. "I've been clear that we need to take this process carefully. We shouldn't rush into triggering Article 50, we should do our preparatory work and be certain about what our objectives are and then start the formal process."

But he said Mrs May would not look upon her new Cabinet as being made up of Remainers and Leavers, but as "a team of Conservatives who have a mandate from the country to deliver not simply Brexit, but the agenda we were elected on a year ago".