That is, how are judges in the UK appointed to cases in general and to this case in particular?
how are judges in the UK appointed to cases in general?
The Resident Judge or the Presiding Judge of the relevant court allocates cases to their fellow judges. Residing Judges sit at specific courts (e.g. all Crown courts will have a Resident Judge). Presiding Judges oversee specific circuits (e.g. the Western circuit).
[how was the judge appointed] to this case in particular?
All extradition cases start off at Westminster Magistrates Court, therefore Vanessa Baraitser, the District Judge assigned to this case, would have been given the case by order of Emma Arbuthnot, the Chief Magistrate of England and Wales.
This is because the Chief Magistrate is the Resident Judge for that court, although in reality I presume she didn't personally make the decision to assign the case. The case listing system may simply have picked Ms. Baraitser as the next available judge to hear this case.