If $G$ and $H$ are groups, then the map $BG\vee BH\to B(G\ast H)$ is a weak equivalence by van Kampen's theorem. However the classifying space $BM$ of a monoid can have arbitrary homotopy type so higher homotopy groups are involved.


If $M$ and $N$ are monoids, is the map $BM\vee BN\to B(M\ast N)$ still a weak equivalence? If there are counterexamples, is there still some criterion that insures such a weak equivalence?



Does the classifying space of monoids commute with wedge sum up to weak equivalence?
Here is a high-tech point of view.


The inclusion functor $\mathrm{Groups}\to \mathrm{Monoids}$, has a left adjoint $F\colon \mathrm{Monoids}\to\mathrm{Groups}$, which is the group completion functor. You know that 
$$\pi\_1(BM) = FM.$$


The claim is that if we instead consider the *total left derived functor* $LF$ of $F$ ("derived group completion"), then we get a sharper result, namely
$$BM \approx B(LF(M))$$
where $\approx$ is weak equivalence. This should apply for any simplicial monoid $M$ (or topological monoid, if you prefer). 


Being a left derived functor, $LF$ must commute with homotopy colimits. If you also know that:


1. The free product of any two discrete monoids is weakly equivalent to their homotopy coproduct as simplicial monoids, and
2. the homotopy theory of simplicial groups is equivalent to the homotopy theory of pointed connected spaces,


then the result follows. 


I think the paper "Simplicial localizations of categories" by Dwyer and Kan (<http://www.ams.org/mathscinet-getitem?mr=579087>) has a nice treatment of these kinds of ideas. (They actually discuss the derived functor of the construction $(C,W)\mapsto C[W^{-1}]$, which associates a category of fractions to a category with a distinguished subcategory. In the case where $W=C$, this is groupoid completion; they show that the classifying space of the derived groupoid completion of $C$ is weakly equivalent to the nerve of $C$.)


**Added:** It appears that the statement you want is proven (in simplicial language) as Proposition 3.8 of the Dwyer-Kan paper I linked to. In that paper, they work with categories which all have the same object set $O$; when $O$ is a singleton, the lemma exactly says that $N(D\*E)\approx N(D)\vee N(E)$, where $N$ is the nerve.


