I doubt that any such records exist (certainly in any organised form). There would seem to be no reason for the British goverment to keep records of workers who'd left the country to toil in the South African mines. Similarly, it's unlikely that the South African government would have kept any official records. So, if your search was to be successful, you'd need to rely upon the records of the mining companies. It's unlikely that many of them would have retained records of people after they'd ceased working for them and, even if they did, very few records would have survived for a century beyond such time.
However it might be worth sending a 'round robin' email to every potentially relevant source on this list. You might just be lucky and find an institution which holds relevant records (or which can point you to another source):
http://www.national.archives.gov.za/reposit.htm