A stamped paper can still be halfway
Official marks are persuasive. A crisp certificate with a seal on it feels like the end of a process, and most people stop worrying once they are holding one. That instinct is what goes wrong here, because the strength of a mark tells you nothing about whether it is the last mark your paper needs. Only the destination decides that.
Why this corridor has a later ending
The Central African Republic has not joined the Hague Apostille Convention, so its offices do not treat the British certificate as a closing act, and embassy legalisation is required before a paper is considered properly legalised there. There is no such finished object as an apostille issued in Birmingham for use in Central African Republic, whatever a search box may suggest.
This is not a quality judgement on your paper. The record can be perfect and recent, and the signature on it entirely genuine. The route is set by which agreements two states have made with each other, and it applies the same way to every sheet, from a company file to a birth record.
Reading your own file, mark by mark
Lay the paper flat and work through what is on it. There should be an issuing body, whose name you already know; a certification of the British signature, done by post from a PO box in Milton Keynes; and finally a consular mark. Name each one and the office behind it. Anything you cannot account for is a gap, not decoration.
Then check what the sheet points at. A closed file names one destination and one purpose, and the chain runs in one direction from the register office outward. If your paper carries marks for a route it is not taking, you are looking at effort spent in the wrong place rather than progress.
The last mark is the one that counts
Getting an apostille from United Kingdom to Central African Republic is not the goal, so do not measure progress by British stamps. Measure it by how close the file is to the consular desk that ends it. That is the stage with the least predictable clock and the one worth reaching first with a complete, correct set of papers.
Start where the paper is made. Birmingham Register Office at Holliday Street issues certified copies of civil records, so the sheet that travels can always be a replaceable one, and the record you keep at home stays out of the chain entirely. Everything after that is timing, counting and patience.