In the UK (and other places like NZ/AU), public servants are permanent. They're not elected in any way.
So they are employees of their departments and don't change when a minister changes. Ministers are almost always a member of Parliament and appointed by the Prime Minister, so they can change at any time.
Usually a government will have some changes at the top of departments (the "Permanent Secretary") and high level executives, but that's also a "change of government to another party" event.
In the US we colloquially call that "career staff". The people at the top are usually political employees, but rank and file will typically stay from one administration to the next.
A big part of the "project 2025" idea was to reduce career employees and make everybody effectively a political appointee.
So they are employees of their departments and don't change when a minister changes. Ministers are almost always a member of Parliament and appointed by the Prime Minister, so they can change at any time.
Usually a government will have some changes at the top of departments (the "Permanent Secretary") and high level executives, but that's also a "change of government to another party" event.