No, there is a body which I think is called the judicial appointments commission.I cant remember who sits on it, but other members of the bar and some politicians are there. It was set up in response to complaints that the judges and senior counsel that did the work previously only ever appointed white men. So now we have, mirabile dictu, female and ethnic minority judges too! It used to work on the “Buggins turn” principle but it is a bit more disciplined these days. Their deliberations, recommendations are still kept under wraps and appointments are made by the Lord Chancellor, who is a member of the cabinet appointed by the PM, and the Justice Secretary.
It is not easy to find senior full time judges because the pay is a fraction of what they earn as senior counsel.
Where The sovereign has a role, it is merely formal and he must act in accordance with the advice of ministers. This is the case for all instances of the Royal prerogative. The sovereign reigns, but does not rule.
PS Ha, I have just thought, the sovereign, of course, does not have sovereignty, that is reserved for Parliament. We fought a Civil war, invited in a Dutch king, and passed a Bill of Rights to ensure that was the case.