The judiciary and the Judicial Appointments Commission have no involvement in politics, and the political parties have no control or influence over them. We don't identify our judiciary as conservative or liberal (or any other shade of political leaning). They aren't appointed by reference to their political views.
Nevertheless, they are one of the key institutions in the constitution. Public, yes - political, no.
This is a bit of a stretch. The important thing to point out is that the JAC has only been in existence for a decade. Prior to this, judges were appointed by the Chancellor, who is a member of the government and an unquestionably political figure. The Chancellor still has veto powers over many of the appointments of JAC including, most relevantly for the comparison with the US, Supreme Court Justices.
There is an attempt to introduce some independent and professional selection process into who gets to be a successful candidate for the position of judge in the UK, so the government cannot just appoint anyone they like, but it's clearly a fiction to pretend judicial selection is completely independent from the government and from political influence.
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In regards the wider, and slightly meaningless, dispute over terminology, we do use the word 'government' differently in the UK. It refers specifically to the executive - the legislature and the judiciary are not considered part of government.