The mechanics of PM appointments in the UK are somewhat different from those in countries with written constitutions. See wikipedia

There is no term of office for a prime minister. The prime minister holds office "at Her Majesty's pleasure". As however to gain supply (control of exchequer funds) that requires that the government be answerable to, and acceptable to, the House of Commons, in reality the convention "at her Majesty's pleasure" means "at the pleasure of the House of Commons". Whenever the office of Prime Minister falls vacant, the Sovereign is responsible for appointing the new incumbent; the appointment is formalised at a ceremony known as Kissing Hands. In accordance with unwritten constitutional conventions, the Sovereign must appoint the individual most likely to maintain the support of the House of Commons: usually, the leader of the party which has a majority in that House. If no party has a majority (an unlikely occurrence, given the United Kingdom's First Past the Post electoral system), two or more groups may form a coalition, whose agreed leader is then appointed Prime Minister. The majority party becomes "Her Majesty's Government," and the next largest party becomes "Her Majesty's Loyal Opposition." The head of the largest Opposition party becomes the Leader of the Opposition and holds the title Leader of Her Majesty's Loyal Opposition. By tradition, before a new Prime Minister can enter 10 Downing Street for the first time as its occupant, he or she is required to announce to the country and the world that he or she has kissed hands with the monarch of the day, and thus has become Prime Minister. This is usually done by saying words to the effect of:

"Her Majesty the Queen [His Majesty the King] has asked me to form an administration and I have accepted."

Although it wasn't required, Tony Blair also said these words after he was re-elected in 2001 and 2005.

Can the last politician to go out the revolving door please turn the lights off?