While the outside of Queen Charlotte's palace in "Bridgerton" is shot at Hampton Court, the interior scenes are mostly filmed at Lancaster House in Westminster, central London. A real-life royal property, owned by the UK government, the house is steeped in royal history. Many of the scenes we see take place there in "Bridgerton", are likely similar to the things the house has seen since it was built in 1825.
However, due to its contemporary usage, it isn't open to the general public. It does, however, host official government events, corporate dos, and more. It can also be seen by the wider public in many famous films and TV shows, including "The Crown", "The King's Speech", and "Sherlock Holmes: A Game of Shadows."