Matching a signature is not reading a document
The treaty slip exists to save a receiving country the trouble of reading foreign paperwork. It says one narrow thing — this signature is real — and every member state agreed in advance to accept that word. Outside the treaty nobody made such a promise, so the receiving side keeps the older habit of looking at the paper for itself.
Where Iran stands
Iran has not joined the Hague Apostille Convention. Britain's own place inside it does not cross the gap, since a treaty shortens a road only where both ends have signed up to it. So a paper heading that way collects the British mark and then meets a reader who was never released from the duty of checking it.
The Wolverhampton to Iran route in practice
The first leg is the same for every country. A single national office covers Britain, works by post, and keeps no counter in the West Midlands. It will mark a register copy, a court copy or a sheet signed by an official it can trace, and it sends the file back the way it came without asking what the file is for.
The second leg is a person with a caseload. Because that reader is doing real work on your document rather than a lookup, the calendar behaves differently: the first stage can be estimated, the second cannot be pinned down from here. Papers tied to a fixed start date should leave this city with weeks to spare.
Which sheet you let travel
A route with two readers and several post rooms is a long chain to put an irreplaceable paper through. A fresh certified copy from Wolverhampton Register Office comes from the same register as the certificate in your drawer and carries the same weight, so send that instead and keep the original where it is safe.