In the UK the decision to prosecute is made by the CPS (Crown Prosecution Service) who are themselves barristers/solicitors on the government payroll - however they do not prosecute the case in court.

The CPS keep a list of approved barristers for criminal cases who they can basically sub-contract the work out to, to actually prosecute the case in court on behalf of the crown.

The cab rank rule basically means those on the approved list arent allowed to pick and choose, the CPS assigns cases to the next in line and they are expected to accept the work irrelevant of the case. Its technically meant to work the same for people looking for a legal professional to defend them, they shouldn't be turned away based on the case, but in actuality its usually fairly easy to find a 'conflict' or a 'diary overlap' if you really don't want to do it.

Its also worth noting a bunch of the people who signed this declaration are not actually barristers on the CPS's approved list - so they couldn't prosecute these cases on behalf of the crown even if they wanted to, its just virtue signalling.