The move that is never wrong
People often freeze at the first question, unsure whether to buy a slip or approach a consulate, and lose weeks to it. The freeze is unnecessary, because the opening move does not depend on the answer. Every road demands a sheet carrying a real official signature, and getting one takes time you can spend now rather than later.
Where Cambodia stands
Cambodia has not joined the Hague Apostille Convention. That single fact decides the shape of everything after the first desk: instead of a slip that closes the matter, the file collects a British mark and then travels on to be read and stamped by staff acting for Cambodia here. The paper you start from, though, is identical.
The Wolverhampton to Cambodia route in practice
The register office at the Civic Centre is the only part of this that is local to you. It holds the civil entries for this city and prints signed copies of them, usually while you wait or within a few days. Everything after that leaves the West Midlands, because one national office covers Britain and works only through the post.
The second leg belongs to the receiving side and behaves accordingly. Consular staff read the document rather than merely matching a signature, keep their own record of what passed the desk, and work to a pace nobody in Britain sets. A file tied to a start date should leave this city with weeks in hand.
Keeping the record you cannot replace
Starting with a certified copy has a second benefit on a route this long. The copy comes from the same register as the certificate in your drawer and carries equal weight, so let it do the travelling through two desks and several post rooms. A lost envelope then costs a reprint charge instead of the only record of a family event.