An independent commission of judges select from ‘proposed candidates of merit’ (15 years service) who then tell the King and PM who they want, who then formalise the appointment.

This is (supposedly) to prevent any particular political leanings, as it’s based on legal experience not political party appointments.

I personally take that with a certain amount of salt, as it helps to have had a certain amount of privilege to get that far - in my personal opinion.

further details: https://www.supremecourt.uk/about/appointments-of-justices.html