There is no counter anywhere in this
Most people picture an office, a queue and a clerk who stamps things while they wait. That picture is wrong from the start here. The stamping office runs by post from a PO box and serves the entire country from one place, so there is no building to visit, no appointment to make and no one to hand a paper to.
Mali's route adds a second postal leg
Mali has not joined the Hague Apostille Convention, so document legalisation for Mali finishes at a consular desk rather than at the British stage. In practice that means a second address, a second wait and a second set of instructions, all handled at a distance in the same way as the first.
There is no such finished item as an apostille issued in Birmingham for use in Mali, so nobody should plan a single round trip. Plan two legs, and expect the second to set the pace, because it is the one you have least visibility over.
What being in the city actually changes
The local part of this job is making the paper, not marking it. Birmingham Register Office at Holliday Street, Birmingham B1 1TJ issues certified copies of birth, marriage and death records, and that is a service with an address, an opening time and a person behind a desk. It is also the only stage where being nearby helps.
After that the job goes national. A sheet leaving the city is treated exactly like a sheet leaving anywhere else, at the same charge and on the same terms. Knowing this stops the search for a local shortcut that does not exist, and puts the effort where it pays.
Planning a job you cannot watch
Work in traceable units. One envelope, one dated list of what is inside, one record of when it went. When several stages run by post, the thing that goes wrong is rarely the stamping; it is losing track of which sheet is where and having no way to reconstruct it.
Build in replaceability rather than optimism. Keep a scan of everything before it leaves, and use certified copies rather than the record you keep at home, so that any envelope in the chain contains something that can be made again. That single habit turns the worst outcome from a catastrophe into an inconvenience.