Current postboxes in the United Kingdom look like this:
Adrian Cable / Hazelwood Hall Postbox / CC BY-SA 2.0
With the "ER II". What will any new ones have on them when Charles is king - will it be CR III?
Current postboxes in the United Kingdom look like this:
Adrian Cable / Hazelwood Hall Postbox / CC BY-SA 2.0
With the "ER II". What will any new ones have on them when Charles is king - will it be CR III?
The "E II R" is known as the royal cypher. In this case, it stands for "Elizabeth II, Regina".
The design of the royal cypher is not fixed; it doesn't have to take the form of Initial — Regnal Number — Title. They can be a bit more elaborate, such as that of Duke Charles III of Brabant, which used three intertwined Cs (while he was the second Charles to become King of Spain, he was the third Charles to become Duke of Brabant).
A few hours after the death of Queen Elizabeth, on the 8th of September 2022, the new King of England has announced that his regnal name will be King Charles III, which puts an end to the speculation (preserved below).
On the 27th of September 2022, his royal cypher was revealed: an intertwined C and R underneath a Tudor crown, with a Roman numeral III in the eye of the R.
Linus Boman offers an in-depth explanation of the design of King Charles III's royal cypher on YouTube.
Also note that current post boxes are not altered; the royal cypher of the current monarch is used on new mail boxes produced during their reign. Over 70 years after his death, some mail boxes bearing King George's royal cypher are still in use today.
What follows is part of my original answer, which was a bit of speculation about the regnal name of the then Prince Charles, prior to the death of Queen Elizabeth.
King Charles
We do not know what regnal name Prince Charles will take when he is crowned King of England. There has been some speculation that, due to negative connotations with the name "King Charles", he might not want to take that name.
Previous Kings Charles
All Kings Charles so far have been Stuarts, while Charles, Prince of Wales, is a Windsor.
King Charles I
King Charles I of England was beheaded for high treason.
King Charles II
King Charles II of England was known for his mistresses and bastard children. About him it was written
Restless he rolls from whore to whore
A merry monarch, scandalous and poor.King Charles III
Bonnie Prince Charlie, had he succeeded in claiming the throne, would've been King Charles III.
Other Options
Prince Charles was christened Charles Philip Arthur George, so he has other names to choose from.
King George
Prince Charles has denied planning to reign as King George VII:
Officially the Prince's office said yesterday: "No decision has been made and it will be made at the time."
"Charles denies planning to reign as King George", The Guardian, 27 December 2005
Which does not state that he will not do so, but just that he is not planning on doing so.
King Arthur
Choosing to become King Arthur would probably be seen as having illusions of grandeur, due to the legacy of that name.
King Philip
King Philip II was a Spanish king who became King of England through his marriage with Queen Mary I. After her death, he waged war on England, a planned invasion failing after much of the Spanish fleet was destroyed. Again a name with some negative connotations.
Conclusion
I think it's safe to assume that of all options, choosing to be King Arthur would be received worst.
So it could become "C III R" for King Charles III, "P I R" or simply "P R" for King Phillip (I) or "G VII R" for King George VII.
The cypher of King Charles III has been announced.
It features his initial C, for Charles, R, for Rex, the Latin for King, and the Roman numeral 3.
It has a Tudor Crown above the letters. There is a separate Scottish cypher which uses the Scottish Crown instead.
As per BBC:
The monogram combines his initial "C" and "R" for Rex, the Latin for king, plus III for the third King Charles.
It was personally chosen by the King, from a range of designs produced by the College of Arms.
… there won't be a sudden change on post boxes … There are still post boxes in use from the reign of Queen Victoria, Edward VII, George V and VI, and the original cyphers remain until boxes need to be replaced.
That symbol, the royal cypher stands for Elizabeth II Regina. This is the made up of the monarch's regnal name, including a regnal number if necessary, and the latin word for king (rex) or queen (regina). A suggestion made by Winston Churchill at the accession of Queen Elizabeth II was that where the numberings differ between England and Scotland the higher be used in both countries, although I'm not sure that there's actually legislation about that. It's even less clear what would happen in the event of Scottish independence.
While Elizabeth II used the first of her given names as her regnal name, there's certainly no reason she had to. For example, her father George VI was christened Albert Frederick Arthur George. So if Charles accedes to the throne he might choose to do so, and thus be Charles III, he might also pick another name. One mischievous suggestion is that he might choose another of his middle names, and rule as King Arthur.