How does the progress of the big bang nucleosynthesis depend on the properties of atomic nuclei?
Nucleosynthesis is dependent upon binding energy.
Every atomic nucleus has a binding energy per nucleon. If the total mass of the component protons and neutrons (=nucleons) is less than the mass of the same number of free protons and neutrons the nucleus is stable and has released energy in its formation.
After Hydrogen, Helium 4 has a very low energy and is incredibly stable. The temperature and pressures when Big Bang Nucleosynthesis occurred produced vast quantities of helium 4 which made it the second most abundant element after Hydrogen.
There is no stable atomic nucleus with 5 or 8 nucleons. This made it impossible for Big Bang Nucleosynthesis to produce any element heavier than the stable Lithium 7 which was only produced in small quantities.