The one job it does
Bristol Register Office, at The Old Council House, Corn Street, Bristol BS1 1JG, issues certified copies of civil records: birth, marriage and death. That is its single relevant function on this route, and it is a useful one. Ordering a fresh copy there means the certificate you keep at home never has to move, and a copy that goes astray can simply be ordered again.
It works by post, so there is no queue in which a translation is examined and no desk where one is approved. Treat the office as a source of clean paper and nothing more. A certified translation into English in Bristol begins after that copy exists, and runs on a completely separate track from the register.
The four jobs it does not do
It does not translate. It does not certify a translation someone else wrote. It does not rule on whether a Zambian form has been filled in correctly. And it does not hold company records, court files or school papers, which live with the bodies that issued them. A certified translation for Zambia touches none of those four functions, so nothing about it is booked through the register office.
This matters most for people who assume certification is always an official stamp. Here it is not. You do not need a sworn English translator in Bristol either, because no court appointment sits behind this route at all. The certification is a signature and a statement, produced by the translator, and it exists the moment the work is signed.
What the English file actually is
A certified translation into English is required for papers used in Zambia, because English is the language Zambian authorities read. Where a Bristol paper is already in English, the language is right and there is nothing to do. Where it is in another language, an official English translation in Bristol is what makes it readable, and that is the only gap the service fills.
The rate is ~£25 / page and the count is the pages you hand over, so the bill is arithmetic you can do before ordering. Every certified translation in Bristol for Zambia comes back as the English text plus the signed certificate of accuracy, delivered in 2–3 business days, or 24 hours on the express route.