Are they affine schemes? It so, you can see Corollary 4.4 here: arxiv.org/pdf/1905.06243.pdf

Actually, you should use Cech overs, not hypercovers, otherwise what you say is only true up to protruncated completion. Hypersheaves on the small etale site of $k$ is not always the colimit you describe, but it is true for ordinary sheaves. See Warning 7.2.2.31 of HTT- it follows that the shape of the hypercompletion of classifying topos of the p-adic integers is not $B \mathbb{Z}_p,$ as these pro-spaces differ on any space all of whose homotopy groups are of $p$-torsion. In the old literature, people were always working up to "#"-equivalence, which is actually just protruncated completion.