In Malaysia, we use the Westminster system akin to UK(ex-colony, you see), and we do have monarch to oversee these process.
1)After the election, the winning party/coalition will submit their preferred candidate as a Prime Minister(can be multiple, but there’s always never more than 1 being submitted).
2)Then the king will summon each party’s leader within the winning coalition and ask them whether the candidate have the party’s support(usually yes else they won’t work together)
3)Then the announcement by the royal palace, and sworn in ceremony.
The whole process, for the past 14 elections, usually will be around 2 day, day one election + ballot counting + result announcement, day 2 PM sworn in. Usually.
The last election(the 15th) kinda break this tradition. It result in hung parliament, where no one coalition win enough seat to be the majority. It took 6 days for the coalition and party of every side to negotiate for a government, the result is the current leading coalition working with ex-arch enemy and chaotic neutral coalition from two autonomous state, leaving a neo-arch enemy and a far right muslim party to be the opposition.