How long does it take to form a new government?
Of course, it depends on the country and it depends on the circumstances.
At one extreme, normally a British General Election is held on a Thursday and, in the course of the following day, a new Prime Minister is appointed by the Queen and immediately starts to appoint a ministerial team.
At the other extreme, under the United States Constitution, the presidential election is held on the first Tuesday after the first Monday in November, but he does not take office until his inauguration at noon on the following 20 January, some two and a half months after his actual election.
In fact, circumstances after the last General Election in Britain in May 2010 were exceptional, since no political party won a majority of seats in the House of Commons, and it took a week to negotiate a coalition government.
In Israel. following the General Election in January 2013, it took two months before a new coalition government was agreed.
In Italy. following the General Election in February 2013, it took two months before a coalition government was agreed.
To the best of my knowledge, the world record is held by the Belgians who took a total of 541 days of negotiations to form the current coalition government (the election was held on 13 June 2010 and the new government agreed on 5 December 2011).