Up to the first 20 minutes, even though the temperature is hot enough for proton-proton (p-p) fusion to occur, there just isn’t enough density for much helium-helium (He-He) fusion, a.k.a the triple alpha process, and no heavier elements to support alpha process or heavy element fusion. The period after this, from S+20 minutes to S+240 kyr (sometimes called the “Photon Epoch” or “Opaque Era”) is dominated by photon pressure which prevents ionized hydrogen and helium ions from capturing electrons, and thus electrostatic repulsion exceeds gravitational attraction and there is virtually no fusion to power nucleosynthesis. At some point in this era matter starts to form the origins of the large scale structures we see today for reasons not fully understood but attributed to weakly interacting “dark matter” that is not affected by light pressure but at ‘local’ scales is too homogenous to form stars and galaxies.
After the universe cooled sufficiently from expansion and becomes transparent, electrons joined with the primordial hydrogen and helium ions to form stable matter that is gravity dominated, and permitted the accumulation of gigantic metal-poor stars with short lifetimes whose hypernovae produced the the bulk of the heavier elements in the universe, specifically carbon, nitrogen, oxygen, which make Earth life possible today, as well as neon, silicon, zinc, nickel, iron, er cetera which allow rocky planetoids to form. It used to be though thst this era of Population III stars only formed several hundred million years after the singularity but several current hypothesis place the origins of heavy matter and stellar firmation further back, possibly starting only a few tens of millions of years after transparency. The bulk of elements heavier than iron, however, had to wait for the second generation of stellar formation and other processes (neutron star collision, cosmic ray spallation, other mechanisms) and the formation of galaxies and galactic clusters to condense matter into tight formations.
While it was previously assumed that most heavy metals were produced directly by supernovae nucleosynthesis, more recent modelling has shown that much of the r-process production occurs in supernova remnants (neutron stars with their associated shock wave and in the accretion discs of black holes) and s-process in asymtopic giant branch stars. So while supernovae produce the necessary conditions for high mass element nucleosynthesis, most of the element production occurs after the actual supernova event, although a lot of elements up to iron are produced by silicon burning as an immediate precursor to core collapse and Type II supernova explosion.