Mrs May, 59, who backed staying in the European Union, has been home secretary since 2010. It was here that she met her husband Philip, a banker - they were reportedly introduced by Benazir Bhutto, later assassinated as Pakistani premier.
"I wish Theresa May every success as she becomes Prime Minister of the United Kingdom and assure her that we stand ready to work with her in the best interests of both Northern Ireland in particular and the United Kingdom as a whole".
"The best thing for the country is having a rapid decision of who the leader should be", agreed Peter Lilley.
Speaking outside Number 10, he said: " She is strong, she is competent, she is more than able to provide the leadership that our country is going to need in the years ahead and she will have my full support".
May, a 59-year-old clergyman's daughter will be Britain's second female prime minister after Margaret Thatcher, who was in office from 1979-1990.
Mr Cameron earlier announced that he will go to the Queen on Wednesday afternoon to offer his resignation. I would also like to pay tribute to David Cameron for the leadership he has shown our party and our country.
And Gove said: "Andrea Leadsom spoke with great dignity and courage today". On Wednesday I will attend the House of Commons for prime minister's questions. The monarch will then invite May - as leader of a party with a majority in Parliament - to lead a new government.
Mr Cameron said he was delighted Ms May will be the next prime minister.
Prime Minister David Cameron says he will step down on Wednesday, making way for Theresa May to succeed him as British leader. "Britain deserves better than this", he said.
"There will be no attempts to remain inside the European Union, no attempts to rejoin it by the back door, and no second referendum".
Theresa May is set to become Britain's next prime minister after her only rival pulled out in a dramatic twist Monday as turmoil sweeps the political scene following the Brexit vote.
Graham Brady MP, chairman of Tory backbench group the 1922 Committee, has explained that both he and the board of the Conservative Party must now formally confirm that Mrs May is the new leader.
He did not suggest re-opening the contest, saying May is now the only contender. The vote has unsettled the markets and sent the value of the pound plunging.
She said her focus was on uniting Britain and, once again, she spelt out that "Brexit means Brexit" - a phrase she has used again and again, which says absolutely nothing about what she thinks Brexit means.
"We now need a new prime minister in place as soon as possible", Leadsom said.
At lunchtime, shortly after a key campaign speech by the Home Secretary, her only rival in the Tory leadership race, Andrea Leadsom, withdrew.
But she has also said Britain should not trigger the exit proceedings until London is ready to start negotiations.
Mrs. Leadsom said in a brief statement that it would be undesirable and detrimental to trigger a highly competitive leadership campaign during such a critical time.
The party is expected to say whether it will to re-open the contest to candidates eliminated in earlier rounds, or declare May the victor unopposed. Currently, it is not known when the new prime minister would be sworn in, However, there are indications that this could happen within a few days.
The announcement comes after Mrs Leadsom apologised to Mrs May "for any hurt I have caused" after the row over comments which appeared to suggest being a mother gave her an advantage as a potential prime minister. May was on her way back from Birmingham after wrapping up her campaign speech.
"Yes ... I feel that being a mum means you have a real stake in the future of our country, a tangible stake".
Despite accusing The Times of "gutter journalism", the newspaper produced an audio and written transcript proving this was exactly what she stated.
That prompted the Times to release a transcript of the interview, with her saying pretty much exactly what the Times reported she had said. Leadsom has children; May does not. She immediately denounced it as "gutter journalism" and the "exact opposite of what I said", but a transcript of the interview set off a fierce weekend debate that left her campaign in tatters. Her allies accused supporters of May - Britain's interior minister - of attempting to undermine Leadsom.
The latest chapter in the political turmoil spawned by the European Union vote moved with breathtaking speed.
Angela Eagle, a senior member of Parliament, officially launched a challenge to leader Jeremy Corbyn on Monday.
The contest was originally scheduled to finish on 9 September.
Many Labour lawmakers believe the staunchly socialist, resolutely uncharismatic Corbyn lacks broad appeal to voters.
Mr Farron, Lib Dem leader, said: "With Theresa May's coronation we need an early general election".