Why the usual certificate is not the one here
Most people arrive at this question having already decided what they want to buy. They have been told a document needs certifying for use abroad, they have heard one word for it, and they treat the rest as an errand. For this destination that order of thinking is the whole problem.
Taiwan has not joined the Hague Apostille Convention. Papers intended for use there require embassy or consular legalisation, and that is the route to plan around from the beginning. Treating the certification as a last formality turns a straightforward errand into two runs through the post, one of which bought nothing.
What the destination's position changes
The single-certificate arrangement works because a group of countries agreed to recognise each other's marks without further checking. Where a destination sits outside it, no British office can close a file alone, however senior that office is. The closing mark has to come from the receiving side's own consular channel, and nothing at this end substitutes for it.
The effect is on how a file is built rather than on what your document says. Nothing about the paper itself changes; what changes is how many desks it passes and which of them has the last word on it. That is a planning question, and it is cheapest to answer first.
The one published charge, and what it buys
The only figures published on the British side are the apostille charges: £45 per document by standard post, and £40 per document on the next-day service open to registered businesses. Each sheet is counted on its own, so several papers mean several charges rather than one.
Those figures are worth naming because they are the ones people budget against by mistake. What you need for an apostille for Taiwan is not really a budget line here, since that certificate is not the one this destination acts on, and the consular channel sets charges of its own.
The Bristol end, which does not change
Whatever the certification turns out to be, the file starts with a sheet somebody official has signed. Bristol Register Office at The Old Council House, Corn Street, Bristol BS1 1JG issues certified copies of births, marriages and deaths, so a civil record can be reissued rather than surrendered.
The FCDO Legalisation Office works from PO Box 6255, Milton Keynes MK10 1XX and serves the whole country by post; there is no counter to visit, in Bristol or elsewhere. That shape matters even where its certificate does not apply, because it explains why this work is measured in envelopes.