I‘d need a reference for the following fact: the one-point union of (nice) aspherical spaces is aspherical. I.e., from $\pi_kX=0$ and $\pi_kY=0$ follows $\pi_k(X\vee Y)=0$. EDIT: Let‘s assume that the spaces are nice, e.g. manifolds.
For the wedge of circles this is true because the universal covering space is contractible.
To answer my own question (in the setting of CW-complexes and thus also for smooth manifolds):
According to Ganea Link to Ganea‘s paper the homotopy fiber of $X\vee Y\to X\times Y$ is homotopy-equivalent to $\Omega X*\Omega Y$ if $X,Y$ are CW-complexes.
If $X,Y$ are aspherical, then their loop spaces are homotopy-equivalent to discrete spaces, hence the join of the loop spaces is homotopy-equivalent to a wedge of circles. In particular, $\Omega X*\Omega Y$ is aspherical. Since $X\times Y$ is aspherical, also $X\vee Y$ must be aspherical.