Historically the first version if the nerve of a covering, which has been used in works of P. S. Aleksandrov in late 1920-s. The nerve of a covering in that version was treated as a simplicial complex had the elements (which are some open sets) of the covering as vertices, and an $n$-simplex is corresponding to $(n+1)$-tuple of elements of the covering which have a common nonempty intersection; in particular one gets a finite combinatorial for a finite covering. This was soon later used in Čech theory. I emphasise this as often nowdays the Vietoris complex which is a bigger complex whose vertices are pairs $(U,x)$ where $U$ is an open set and $x\in U$ is nowdays often called Čech complex as well, as the finite, original, version is now more rarely used. Simplicial sets replaced old-fashioned simplicial complexes a couple of decades later.

Grothendieck generalized the nerve to the case of categories. The simplicial complexes in their combinatorial and topological reincarnation were from the beginning taken interchangeably. However, for simplicial sets, nice categorical treatment is from Milnor, who formally introduced a notion of geometric realization in modern context; the concept was essentially known but not its properties at the time. Classifying spaces for group case, were of course studied first in the context of group cohomology, so MacLane is probably among the first ones using it. Segal in late 1960s, not only studied the concept in depth but also introduced more complicated version for simplicial categories.

Historically the first version is the nerve of a covering, which has been used in the works of P. S. Aleksandrov in late 1920-s. The nerve of a covering in that version was treated as a simplicial complex had the elements (which are some open sets) of the covering as vertices, and an $n$-simplex is corresponding to an $(n+1)$-tuple of elements of the covering which have a common nonempty intersection; in particular one gets a finite combplex for a finite covering. This version was soon later used in Čech theory. I emphasize this as often nowdays the Vietoris complex which is a bigger complex whose vertices are pairs $(U,x)$ where $U$ is an open set and $x\in U$ is nowdays often called Čech complex as well, as the finite, original, version is now more rarely used. Simplicial sets replaced old-fashioned simplicial complexes a couple of decades later.

Grothendieck generalized the nerve to the case of categories. The simplicial complexes in their combinatorial and topological reincarnation were from the beginning taken interchangeably. However, for simplicial sets, the nice categorical treatment is from Milnor, who formally introduced a notion of geometric realization in modern context; the concept was essentially known but not its properties at the time. Classifying spaces for group case, were of course studied first in the context of group cohomology, so MacLane is probably among the first ones using it. Segal in late 1960s, not only studied the concept in depth but also introduced more complicated version for simplicial categories.