The issuing state does the marking
An apostille is a slip that one state fixes to its own public papers so another state will read them without a further argument. That shape explains the rule people trip over here. Britain marks British papers, and a police certificate drawn up anywhere else remains that country's paper and that country's business. No office in Milton Keynes has any standing over a sheet a foreign force signed.
The British certificate's road
To apostille a Police Clearance Certificate in Wolverhampton, nothing at all happens in the city itself. The sheet travels by post to the FCDO Legalisation Office, PO Box 6255, Milton Keynes MK10 1XX, which does this work for everyone in the country. Britain has been inside the Hague Apostille Convention since 1965, so the slip it issues is a long-settled item and not some new arrangement a reader abroad will puzzle over.
Price and pace
The standard postal charge is £45 per document, and a registered business can pay £40 per document for the next-day turn. That figure attaches to each sheet. So a person holding two certificates from two countries is not paying it twice here, because the foreign sheet is not a job this office can take on at all. Only the British one belongs in the envelope.
Common trip-ups
The commonest slip is assuming one envelope can carry everything a move abroad calls for. A certificate obtained in another country cannot be swept in with the British sheet; it comes back untouched, and the lost fortnight lands at the worst possible moment. The second is starting before the certificate has arrived, which moves you along the queue without shortening it.