The postmark used by the british royals was redesigned after Queen Elizabeth II's death to mark the new reign. As part of the redesign, the Royals switched their PPI number back to W7047 after using W4047 since 2019. This postmark was only used for about a year, because being redesigned due to new franking machine and Royal Mail's switch to Mailmark machines.
This blog post covers all the postmarks I received that feature the this designs. You can find info on the past & the new postmark used here.
Court Post Office at Buckingham Palace
The Court Post Office at Buckingham Palace processes mail to and from Buckingham Palace, Clarence House, St. James Palace, Kensington Palace, and occasionally other Royal Residences outside of London. The following shows the same postmark from the Court Post Office presented in different circumstances.
Air Mail Stamped
Air Mail Sticker
Double Postmark
I did not receive this reply during this era.
2nd class
I did not receive this reply during this era.
Court Post Office at Windsor Castle
The other year-round Court Post Office is at Windsor Castle, which handles mail for the Royal Residences in that area (Windsor Castle, Royal Lodge, etc.). Although I have seen people get mail from Windsor Castle with the Buckingham Palace postmark.
The other Royal Residences don't have their own full-time Post Office. I'm not entirely sure on the details on how their mail is processed. Some of them will just have their mail taken to London and dealt with the Court Post Office at Buckingham Palace. Others have their own Franking Machine to stamp the envelopes, and then they will just deliver their mail to their local post office.
Sandringham House
This postmark is used for Sandringham House. King Charles spends Christmas and part of his Winter holiday at Sandringham House. I know during Queen Elizabeth's reign, Royal Mail set up a temporary post office on the Sandringham estate when the Queen was in residence.I did not receive this reply during this era.
Other Royal Residences
This postmark is for Bagshot Park, the Wessex's residence and office. It has in the past been used for other royal residences, which is why it is not specific to Bagshot Park.