I asked this question in /r/history because I'm really interested in who (if anyone) will declare, "The Queen is dead! Long live the King!" and I understand that this would break the year rule here. How has it historically been done?
Also, could someone speak to what will happen in the eventual passing of the queen in the UK, who will have this honor/responsibility? Will it still be done? Will it be televised? Is it delegated to a position, or is the task chosen at the moment?
I do not think this is how the phrase is meant. It's not an actual proclamation that the king is dead, like someone has to say it when the king dies for it to be official (if they don't, would he still be alive?). Rather, it's an affirmation of the system of governance in which there is ALWAYS a king. Whenever the king dies, the kingship - the office of being king - automatically moves to the heir.
This is tied together with the notion of the Kings Two Bodies, which you might also find interesting. The TLDR version is that one body is the king's physical body, while the other is the body politic. That idea comes from the notion of treating the nation as a human body, which is manifested in the very person of the monarch. Perhaps the most famous display of this notion is when Louis XIV of France proclaimed, "I am the state".
Medieval and early modern poli sci is really kind of fun stuff.
In France the statement was made by the senior Duke, the Duc d'Uzes. The format was to repeat the intial statement three times, for example "Le roi Louis XIV est mort. Le roi Louis XIV est mort, Le roi Louis XIV est mort" then break a stick to symbolise the finality of the rule, then joyously proclaim "Vive Louis XV!".
The official proclamation will be made by heralds, at least in London- here's the present Queen's accession being proclaimed at the Royal Exchange in London by (the comments tell me) Clarenceux King of Arms. And yes, this was televised live at the time, it was a proclamation to all the Queen's subjects after all.
The proclamation is actually drawn up by an Accession Council that meets as soon as possible after the previous monarch's death- here's what the Monarchy's official website says about it. In 1953 anyway, copies were sent to every Borough in the land- I once came across Aylesbury's copy misfiled with something quite unrelated.Rather disappointingly it was just a printed sheet, with as I recall the text as read above and the names of all the Councillors present.