well OK- this is clearly an issue for the British Authorities, so I am not sure if the best answer - try South Africa house, Trafalgar sq really is that good.
Basically it is the jus soli ( law of the land ) that governs nationality rather than jus gentium ( law of the people ). The children should have been registered at birth with the British High Commission in SA. I cant imagine a Brit ex-pat didnt tell him at the time.....
The current Act is the Nationality Act 1981 and is here:
http://en.wikipedia.org/wiki/British_Nationality_Act_1981
At one time I was looking at this for ( actually unbelievably ) a South African relative, He didnt have British nationality himself, but somewhere ( unless it has been repealed) there is a "grandfather clause" which allows the grandchildren to claim a british passport BUT they are then NOT able to give/pass on nationality rights to THEIR children ( the gt grandchildren ).
Having done your reading I think you are going to have to take legal advice ( and pay for it ) and you can thank yourself that at least you are not trying to sort out relatives from India under the British Raj ( those rules are terribly confusing )