No, he or she does not have a public ceremony, highlate is pretty much on the mark, although the image of the Monarch saying, " get on with it ", is an amusing one to imagine, I think that he, or she, would phrase it a little more delicately than that.
Although, in practice, that is pretty much what the inaugural procedure boils down to.
There is a type of ceremony held privately with the reigning monarch.
The British Prime Minister does not occupy his or her position by virtue of their party winning an election, the Prime Minister occupies that office because he or she is appointed by the reigning monarch.
The formal title of the British Government is, Her, ( or His ), Majesty's Government, because the Monarch is the head of state, the Prime Minister isn't.
The prime Minister, and every other minister, only occupy their offices, and are permitted to form a government, at the monarch's pleasure.
There is no such thing constitutionally, as the British people's government, there is only the monarchs government, although in practice, the will of the British people as expressed by their electoral votes is what governs the choice of which party forms the government and of who is appointed as Prime Minister.
Constitutionally, the monarch can select as Prime Minister, any member of parliament that he or she wishes.
In practice of course, the monarch always selects the leader of the political party which has the majority of candidates elected as members of parliament in the election.
However, if the parties with most members of parliament had an equal number of members of parliament, the monarch would have to select the leader of one of those parties to be Prime Minister, unless one of the parties could gain a majority of seats in parliament by forming a coalition with a smaller party, in that case, the monarch would appoint as Prime Minister, the leader of the larger party in the coalition.
Also, if a Prime Minister resigns from office, his party will select a replacement and the monarch will invite that person to become Prime Minister, but if the party can't decide on a candidate, the monarch, after taking advice, will pick someone from that party to appoint as Prime Minister.
In practice, after a general election, the monarch summons the leader of the party with most members of parliament to Buckingham Palace, the monarch then " invites ", that person to form a new government, if they accept the invitation, it's at that point that they become Prime Minister and First Lord of the Treasury, which is an additional office that is automatically occupied by the Prime Minister.
Next, the newly appointed prime Minister kisses the monarch's hand, which is why the Prime Ministerial appointment procedure is known as, kissing hands, or, the kissing of hands.
Next , either immediately, or as soon as possible afterwards, the Prime Minister has to swear an oath of loyalty to the monarch.
That's done either in the presence of the monarch, or not, as the monarch commands.
Actually there are two oaths sworn, the Oath of Allegiance and the Official Oath.
The oaths are administered by a senior civil servant who occupies the post of Clerk of the Council, the council in question being the Privy Council, which is is an advisory body to the monarch, and is made up of senior politicians, privy means private, and because privy means private, in Britain the word privy is also a euphemism/slang expression for a toilet. !
The Prime Minister doesn't even swear the oaths as Prime Minister, he or swears the oaths as First Lord of the Treasury, of course a Lord is always a male, and a Lady is a female, but a female Prime Minister would still be called First Lord of the Treasury.
The Oath of Allegiance and the Official Oath, as set out in the Promissory Oaths Act 1868 are required to be taken by various office-holders.
The Oath of Allegiance is in the following form:
I, NAME, do swear that I will be faithful and bear true allegiance to Her Majesty Queen Elizabeth, her heirs and successors, according to law. So help me God.
The Official Oath is in the following form:
I, NAME, do swear that I will well and truly serve Her Majesty Queen Elizabeth in the office of .... So help me God.
Various other people such as newly appointed government ministers also have to swear these oaths.
No one in Britain ever swears oaths of loyalty and obedience to the government, country, or people, it's always to the monarch.
For example, people joining the armed forces swear allegiance to the monarch, because the monarch is the commander in chief, although in practice, that role would be carried out by the Prime Minister, but only because the monarch grants those powers to the Prime MInister.
Job done, and it's off to the Prime Minister's official residence at 10, Downing Street for tea and buns, or something stronger, and if the Prime Minister is a new one, they will have arrived at the Palace in their own vehicle, but will leave in the Prime Minister's official limousine, complete with police escort, same thing in reverse if they resign from office, they resign to the monarch, and instantly lose all Prime Ministerial privileges, such as the use of the limo