Despite the number of ATCOs of the old school who think of the job as blue-collar and regard graduates as having no common sense, the simple fact is that it's becoming more and more a graduate profession. A degree may not be particularly helpful in the early stages of an ATC career, but if you hope one day to climb the greasy pole (the old salts won't like that either; ignore them!) it's likely to be more helpful than not.
A degree in aeronautical systems/engineering or the like may be useful in recruitment terms (it is evidence of a genuine interest in aviation), but only marginally. A PPL would also display that interest, as would a history of visiting towers and centres. My advice would be to undertake whatever degree subject most interests
you, and if it happens to open up a fall-back position then so much the better. If aeronautical systems is that subject, go for it.
As for where to apply, NATS and Eurocontrol are the obvious ones. To apply to Nav Canada, ATNS or Airservices Australia for ab-initio training, you'll need to already have the right to live and work in Canada/SA/Aus before starting training. The number of South African controllers living and working elsewhere, the Middle East in particular, would suggest that SA is not the place to be anyway! From what I've gathered, ATCO salaries in SA might get you a decent standard of living but are still low in international terms. Crime is a huge issue in SA, and if you have kids in the future you may need to find the money for private education. Industrial relations are not good, from what I've heard. Nav Canada salaries are pretty decent, and will generally give a better standard of living than the UK thanks to lower living costs, though Canadians get less leave than Brits and some centres and towers rely heavily on overtime. Australia has significant industrial relations issues (try a search on this forum), and both Aus and SA tax the **** out of the standard ATCO toy, the nice imported car... The Commonwealth countries no longer give any preference to immigration from the UK, so frankly it's likely to be a non-starter anyway to consider those countries unless you have parents/siblings already there.
In summation, a degree is likely to be helpful, and there's nothing to stop you applying to NATS/Eurocontrol while still at university. Be realistic in your options (as in, if you want to be overseas it may have to become a long-term goal) and keep a Plan B open in view of the high failure rate in ATCO training.
Good luck!