What functor does $K(G, 1)$ represent for nonabelian $G$?

K(G,1) aka BG classifies G-bundles — i.e. G-coverings, if G is discrete. (Details can be found e.g. in May's Concise Course in Algebraic Topology.)

Usual definition of Cech cohomology works for $H^1(X;G)$ even in non-abelian case (but it's just the usual cocycle definition of G-bundle).

As for universal coefficient theorem, even if $H_1(X;\mathbb Z)$ is trivial, $H^1(X;G)$ needn't be; but (if G is discrete) $H^1(X;G)=\operatorname{Hom}(\pi_1(X);G)/\text{conjugation}$ (reference: Hatcher, 1B.9). (But if one wishes to consider BG for general G, things get worse — "$H^1$" is no longer defined by 2-skeleton of X. Perhaps, AHSS from cohomology to K-theory can be viewed as kind of "universal coefficient spectral sequence" for $G=U=\lim U(n)$.)