[quote=@Mihndar]
But at least until World War 3, the Nuclear Non-Proliferation treaty and the general trend of less nuclear weapons and developing countries not having the technology would have held in place. Other countries don't have nuclear weapons because of technology, they don't have them because they didn't want to build them and furthermore weren't allowed to. South Africa had nuclear weapons, but they disarmed them and ended their nuclear program due to international pressure. So until World War 3 actually started and the structure of international law fragmented, no nation could have a nuclear program without the treatment that Iran and North Korea are currently subject to, namely massive, economy-crushing sanctions. Regardless of whether or not you have the technology (which would still be a highly guarded secret and would take a few years to work out, as Iran and North Korea have been doing for a decade), it takes a while to refine the uranium required and actually put it into practice as a warhead. There is no way a country like Ethiopia could have nuclear weapons yet. In a few years, sure.
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Yeah, but this is thirty-five years in the future. It's probable that the rate of nuclear proliferation was flung into high gear during the onset of WWIII. As the war gradually intensified, and as nuclear weapons were utilized, this proliferation rate might've been dramatically energized. 

And if you're a superpower and you're still using uranium to produce nukes, then you've failed to keep pace with your aspiring adversaries. You should be producing pure fusion, antimatter-catalyzed fusion, or pure antimatter warheads at this point. 

Ethiopia should at least be able to drum up a few gun-type fission bombs. They don't require plutonium, are comparatively cheap to produce, and don't require a robust technological base to engineer and manufacture--but they're woefully inefficient when compared to more sophisticated nuclear weapon designs. Implosion-type nuclear bombs are an option as well, but they're a fair bit more advanced than their gun-type cousins.