Well I can give you the info I have as I don't think much has changed since I applied.
We went through Breytenbach attorneys in the UK but they also have an office here in SA in Pretoria.
There are basically three ways of getting citizenship for children in the UK:
1) One of your parents were physically born there (i.e. they have a UK birth certificate). You provide this as proof and you get a passport after attending the necessary ceremony.
2) One of your grand parents were born there. You get an ancestral visa. I am not sure of the requirements for this though as it wasn't applicable to me.
3) One of your parents have a british passport but were not born in the UK. This applied to me and my father was instructed to get me to the UK as soon as possible to apply for a permanent residence visa. This entitled me to live and work in the UK until such a time I decided to apply for citizenship.
I could apply for my passport after I had been resident in the UK for 5 years, during this 5 year period I was not absent from the UK in total for more than 450 days and during each year, not more than 90 days.
Once I have done that, I could apply for my british passport.
The catch was that I had to be there in the UK physically to apply for this visa, I had to be younger than 18 (the further away from 18 the better my chances of getting the visa as the attorneys had said that I had a very slim chance of being granted this visa if I was 18 exactly) and my dad had to prove that he could support me.
I went over when I was 16 and stayed for a year as it took about 6 months in total to get this visa and it cost a pretty penny. After that year, I came back to study at University to get my degree I wanted. I am in the final year of my degree and will be going back to start the 5 year process as soon as I am qualified as a CA(SA). The final catch is that I could not be outside the UK for a consistent period of 2 years. So I have gone back to England every year to ensure this visa stays valid until my return.
Those are all the details I know of and that is reading and writing straight from the letter from the attorneys I hauled out to inform every one on this thread. No matter what anyone else tells you, these are the only methods of getting British citizenship short of marrying a Brit.