While I agree with the first part of @MatthewHerbst's answer, I strongly disagree with his suggestion to "lose" your passport, so take this as a very extended comment if you wish:
It depends a bit on your nationality whether you need a Schengen / UK visa or are eligible for a visa / leave to remain on arrival.
However in both cases either in the visa application and/or at immigration you might be asked whether you had previously overstayed or have or had a ban from some country. It is a very bad idea to lie in this context.
While "losing" a passport might initially sound like a smart idea especially in the case that you do not need to apply for a visa, I highly recommend against it. Losing your passport means there will be a police report/flag somewhere about a lost passport and security agencies are sensitive to this, as a similar strategy is also used by "terror tourists" to camps in Afghanistan/Pakistan/Syria or wherever else the world is on fire right now. There is a chance that immigration is getting wind of this and will give you extra scrutiny. At this point I also would not at all be surprised if the authorities of South Africa and the UK but also Schengen shared access to their databases and already knew about your overstay and now will be very curious as to why you "lost" your passport so recently.
Let me conclude by citing from this excellent answer on another question, citing UK procedures:
The strategy of concealing an adverse immigration event by 'losing' one's passport and getting a fresh, unblemished passport is a poor one. There is a history associated with the passport that is not accounted for in its physical pages, but rather in computer systems linked to the passport number. And a new passport will contain a record linking it back to the previous one. People still try this strategy, however, and when they get caught the results are catastrophic.
If the UK catches somebody doing this, they will get logged (along with their biometrics) as violating Paragraph 320 of the Immigration Rules. That usually means the person can forget about coming here for a long time, if ever. Plus they will tell the US Department of Homeland Security about it (regardless of your nationality). They keep your biometrics on file for 12 years if you have a clean history, and indefinitely if you fall under Paragraph 320.