Nobody passes the file along for you
It is natural to imagine offices handing a document to each other until it reaches its destination. That is not how this works. The British stage begins and ends with the person who posted the sheet in, and the marked bundle returns to that same address. The chain has a gap in the middle, and you are standing in it.
That gap is the practical difference between this corridor and one where a single mark finishes the job. On those routes the marked sheet can go abroad the moment it lands. Here it lands, waits on your table, and then starts a second journey that only you can set in motion.
Why Taiwan needs the second mark
Taiwan has not joined the Hague Apostille Convention. Where two countries share that treaty, each accepts the other's certificates as they stand; where they do not, the older arrangement applies and an officer acting for the destination looks at the file directly. So embassy legalisation follows the British stage rather than replacing it.
Using the time the file is with you
The days between the two journeys are useful rather than dead. This is when errors are cheap to find: a middle name spelt differently from the one on your passport, a date written the American way round, a sheet that turns out to be the short version of a certificate when the long one was wanted. All of that is fixable at home.
It is also the moment to make sure the bundle stays whole. The certificate is fastened to the page it belongs to, and the pair travel as one object from here on. A loose slip and a loose sheet arriving together at a consular desk are not the same thing as a bundle, and separating them creates work nobody needs.
Where the route starts in Cardiff
Cardiff Register Office at City Hall, Cathays Park reissues birth, marriage and death records, and each copy carries a registrar's signature the British stage can work from. The FCDO Legalisation Office then handles the marking for the whole country by post, so nothing about this route involves a counter in Wales or an appointment anywhere.