What joining the Convention changed
The Convention works by agreement: states inside it accept each other's slip instead of running their own checks. Saudi Arabia came into that agreement in 2022. Britain was already inside it, and from that point the corridor between the two had both ends covered, which is what turns a chain of desks into a single mark.
What 'sufficient' means on a form
Sufficient means nobody adds anything after the British office does. There is no consular queue to join, no second charge to budget for and no third body reading the file. Document legalisation for Saudi Arabia is done in one stage, which also makes the timing easier: the clock is a postal one and you can read it.
It also means the finished bundle should stay whole. The mark is fixed to the sheet it speaks for, so keep the two together and photograph them for your own records rather than taking the bundle apart to file the pages separately.
The office that issues it
One national office issues every British mark, wherever the paper started life, and it works through the post rather than over a counter. A paper from a London hospital, a London court or a London register goes to the same address as one from anywhere else. What makes the London part of the job real is the register office that prints your copy, not a local marking desk.
Getting it right the first time
Start with a version of the paper that a public body signed, because an unsigned printout gives the office nothing to work with. A fresh certified copy of a birth, marriage or death record from Westminster Register Office does the job and keeps your own certificate safely out of the post.
After that an apostille from United Kingdom to Saudi Arabia is a short, predictable errand. In practice, what you need for an apostille for Saudi Arabia is one signed public paper, one fee, and one trip through the post in each direction.