I love this question! Really, the FCO should include "Closest local equivalent to Greggs" alongside its safety and visa information for every country. It's essential information :-)
There are quite a lot of Greggs-like pastry-heavy bakeries in South Africa. One widespread franchise is King Pie, who are quite Greggs-like: (pic from WikiVillage)
Their "steak pie" pasty should cure any Greggs addict's Steak Bake withdrawal, and the Steak and Kidney pasty is possibly more Greggs than Greggs itself.
FYI, in South Africa, a "Russian" (like in "Russian roll" in this image) is basically a large frankfurter-type hot dog sausage. Apart from that it's all very familiar, but if you want a cuppa with your pasty, ask for "Ceylon tea" or "Five Roses" as "tea" often defaults to rooibos and "black tea" sometimes causes confusion (on occasion in KZN I asked for black tea and they didn't recognise the phrase and gave me black coffee instead - I suppose they thought "black tea" must be like the opposite of "red cappuccino").
There are also many stands serving Greggs-like recently baked, rapidly cooling pasties in the shops next to petrol stations on main motorways. I didn't catch the names, but at least one "Bonjour" shop at a Total station had one of these.
Finally, you can also get freshly baked pasties at some large supermarkets' in-store bakeries, including Woolworths, which in South Africa is an upmarket Marks & Spencer-like supermarket/department store chain.
South African Woolworths is a bit more swanky than Greggs though: (image source)