The word has escaped its own meaning
'Apostille' began as the name of one specific slip issued under a treaty. It is now used abroad as a general term for making a paper official, in much the same way people say 'hoover' for any vacuum. That is harmless until the destination sits outside the treaty, at which point the word and the actual route stop matching.
What applies on this corridor
South Sudan has not joined the Hague Apostille Convention. Document legalisation for South Sudan therefore runs the older way: the British mark first, then a stamp from the consular office that acts for that country here. An apostille from United Kingdom to South Sudan is a real and useful mark, but on this route it is a step rather than a finish.
None of that makes the British stage optional. The second office is looking at a bundle the first has already vouched for, so both belong in the plan from day one. What changes is where you set your expectations: the day the envelope returns from Milton Keynes is a milestone, not the end.
Read the request as a result, not an instruction
Take the letter you were given as a description of what has to be true at the other end. If it says the paper must be recognised in South Sudan, that is the part worth acting on. If it names a mark by name, treat the name as shorthand and follow whichever route actually reaches that country.
This matters most when a date is attached. Someone who reads the word literally books a single postal job and nothing else, then finds a further office waiting when the deadline is already near. Reading it as shorthand puts that office in the plan at the start, which is the cheapest correction anyone on this route can make.
The half you can pin down today
Start with the British half, because its price and its steps are published. Westminster Register Office at 317 Harrow Road issues a fresh certified copy of a birth, marriage or death record, so your own certificate stays at home. The Legalisation Office charges £45 per document by post, or £40 per document on its next-day line for a registered business. That is what you need for an apostille for South Sudan at the British stage; the second office sets the rest.