The barristers the CPS employs to bring prosecutions are the same barristers used by the Post Office, using the same courts and the same judges.
That’s actually not entirely true. Although the CPS does engage “free” barristers via chambers for some cases, most CPS prosecutions are handled “in house” by salaried barristers working directly for the CPS.
CPS’s in-house barristers are (as a rough rule) extremely experienced at prosecuting common-or-garden cases, but lack the specialist experience of barristers available to hire via chambers, who they will usually bring in for the more complex prosecutions (or ones involving a specialist area of expertise).
All barristers are only as good as the evidence given to them, though, and one of the real strengths of the CPS barristers is experience in working with the police- both in terms of knowing how to get the best evidence out of them, and knowing a police wild goose chase when they see one. This is the part that really breaks down in cases like the Post Office, where it’s private corporate investigators throwing complex technical evidence over the fence at random barristers who have mostly not worked with them before.