Start from the date, not the paperwork
Most trouble on this route comes from starting at the beginning. A deadline-led plan begins with the day the page must be in front of its reader and adds each stage backwards from there. Doing it that way exposes the real constraint quickly, and the real constraint is almost never the stamping office. It is whoever has to release a signed version of the paper in the first place.
The stage nobody can hurry
Edinburgh Registration Service prints certified extracts of Scottish births, marriages and deaths from City Chambers, and it prints them on its own schedule. Universities, courts and companies are the same. Order early, order more than one copy if the matter is important, and let the fresh extract travel so the original never leaves you. A spare extract is cheap insurance against a second full run.
The stage you can actually plan
The FCDO Legalisation Office at PO Box 6255, Milton Keynes MK10 1XX handles the whole country by post. It has no counter, so the timing is postal in and postal out with a short clerical step between. Because an apostille from United Kingdom to Nicaragua needs no second desk, this middle stage is the one part of your timetable with a shape you can rely on.
What speed costs
Standard postal handling is £45 for each paper. The next-day route, open to registered businesses, is £40 for each paper. Buying speed at the middle is therefore cheap and buying it at the ends is not, because tracked international carriage and rush translation both carry real premiums. Spend the money where the calendar is tightest rather than spreading it evenly.
The last stretch into Nicaragua
Nicaraguan offices work in Spanish, so a rendering of both the paper and the slip is a common final task, and it belongs after the stamp rather than before it. Leave a genuine gap for it. A timetable that ends the moment the stamped page lands back in Edinburgh is a timetable that stops one stage short of the reader.