For $p=2$ there are references from which you can extract the $\mathbb{Z}/2$ cohomology algebra of $B(M,2)$ for any closed manifold $M$. The answer depends only on $H^\ast(M;\mathbb{Z}/2)$ as a module over the Steenrod algebra, but is usually not very pleasant (even when this module is). The method consists of showing that the map $$ B(M,2)\to S^\infty \times_{\mathbb{Z}/2} M\times M $$ is surjective in cohomology, and calculating its kernel. The situation is nicely summarised in Section 4 of

Bausum, David R. Embeddings and immersions of manifolds in Euclidean space. Trans. Amer. Math. Soc. 213 (1975), 263–303,

where you will find references to the original articles of Haefliger and Yo Ging-Tzung.

For $p=2$ there are references from which you can extract the $\mathbb{Z}/2$ cohomology algebra of $B(M,2)$ for any closed manifold $M$. The answer in principle depends only on $H^\ast(M;\mathbb{Z}/2)$ as a module over the Steenrod algebra, together with the Stiefel-Whitney classes of $M$, but is usually not very pleasant (even for manifolds with nice cohomology such as $\mathbb{R}P^n$). The method consists of showing that the map $$ B(M,2)\to S^\infty \times_{\mathbb{Z}/2} M\times M $$ is surjective in cohomology, and calculating its kernel. The situation is nicely summarised in Section 4 of

Bausum, David R. Embeddings and immersions of manifolds in Euclidean space. Trans. Amer. Math. Soc. 213 (1975), 263–303,

where you will find references to the original articles of Haefliger and Yo Ging-Tzung.