In the USA it is the Chief Justice of the United States. In Russia it is the Chairman or the Central Electoral Commission. In Byzantine Empire it was the Patriarch. What are other examples? What are the patterns?
In the same way the Holy Roman Emperor was for a time an elective office. The Elector of Hanover for example, who then became King of England. Originally the HRE was an autocracy founded by the Carolingians, then evolved into an electorate then slipped in and out of autocracy and elections. Partly because the emperor was catholic and some of the German principalities/electorates became protestant, the Hapsburgs largely made it hereditary, and it was Napoleon who abolished it as Alex says.
For the UK and Commonwealth monarchies, the new monarch succeeds immediately and automatically on the death or abdication of the previous one. This means that it is possible to be monarch while unaware of the fact. This happened to the current queen, who was in Kenya at the time of her father's death, and was not aware of her accession until the news reached her. So there isn't any one person who inaugurates the monarch.
The monarch is required to swear some oaths as soon as possible in the presence of the Accession Council. This is a body made up of persons who have been appointed to the Privy Council by previous monarchs, mostly politicians, but including other significant people, plus members of the House of Lords, the Great officers of State (which are now largely ceremonial positions), the Lord Mayor and Aldermen of the City of London, and the High Commissioners (equivalent to Ambassadors) of the Commonwealth Realms. The new monarch is required to take these oaths; their only way to avoid doing so is to abdicate.
The formal coronation of the monarch is largely a celebration, without much legal or constitutional importance, and is usually held about a year after their succession - it takes a lot of organising.
For the papacy, which is an electoral post, when a ballot has produced a positive result, the man elected has a chance to decline it, but if he accepts and is already a bishop, he becomes pope immediately. If he isn't a bishop, he first needs to be consecrated as one, and then becomes pope immediately that's done. The question, and if necessary, the consecration are administered by the Dean of the College of Cardinals, or if he isn't available, the sub-Dean, his deputy. If neither of them are available, the senior Cardinal Bishop on hand does it.