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The nucleons that date back to the stage of the starting of the universe were made in the same procedure.This was when the Big Bang contained quark-glucon plasma which cooled the temperature below approximately 10 million degree Celsius.
As it expanded, it cooled, and the quark-gluon plasma ‘froze’ into neutrons and protons (and other hadrons, but their role in BBN was marginal), which interacted furiously … The universe continued to cool, and soon became too cold for any further nuclear reactions …
the unstable isotopes left then decayed, as did the neutrons not already in some nucleus or other.
This ‘r process’, as it is called (actually there’s more than one) produces most of the elements heavier than the iron group (copper to uranium), directly or by radioactive decay of unstable isotopes produced directly. Here are a few links that might interest you: Nucleosynthesis (NASA’s Cosmicopia), Big Bang Nucleosynthesis (Martin White, University of California, Berkeley), and Stellar Nucleosynthesis (Ohio University).
Plenty of Universe Today stories on this topic too; for example Stars at Milky Way Core ‘Exhale’ Carbon, Oxygen, Astronomers Simulate the First Stars Formed After the Big Bang, and Neutron Stars Have Crusts of Super-Steel.
During the subsequent expansion, this plasma has progressively cooled down.
By examining how the cooling affects the matter content of the universe, one can derive one of the most impressive testable predictions of the big bang models.
The big bang models - the cosmological models based on general relativity - tell us that the early universe was extremely hot and dense.
At the earliest stages that can be modelled using current physical theories, the universe was filled with radiation and elementary particles - a hot plasma in which energy was distributed evenly.
This produces most of the lithium (apart from the BBN Li), beryllium, and boron.
And one more: in a supernova, especially a core collapse supernova, huge quantities of new nuclei are synthesized, very quickly, in the nuclear reactions triggered by the flood of neutrons.