Here are some details which are related to Tyler's comment.
I recommend looking at the paper "Induced Fibrations and Cofibrations" by Tudor Ganea (1967). For connected based spaces $X$ and $Y$, there is a fibration up to homotopy $$ \Sigma (\Omega X) \wedge (\Omega Y) \to X\vee Y \to X\times Y $$ where the first map in the display is a kind of generalized Whitehead product (see below).
After looping once, the sequence splits, so $$ \Omega (X\vee Y) \simeq \Omega X \times \Omega Y \times \Omega \Sigma ((\Omega X) \wedge (\Omega Y))\, . $$ Your isomorphism will follow by applying $\pi_{n+k}$ to this splitting--we only need to identify the term on the right.
To this end, not that if $X$ is $n$-connected and $Y$ is $k$-connected ($n,k\ge 1$), then $\Omega \Sigma ((\Omega X)\wedge (\Omega Y))$ is $(n-k-1)$-connected (here I am using the Hurewicz theorem). Moreover, the map $$ ((\Omega X)\wedge (\Omega Y))\to \Omega \Sigma (\Omega X)\wedge (\Omega Y) $$ is $(2n+2k-1)$-connected. In particular, it will induce an isomorphism on $\pi_{n+k}$.
As $((\Omega X)\wedge (\Omega Y))$ is $(n+k-1)$-connected, the Hurewicz theorem says that $$ \pi_{n+k} ((\Omega X)\wedge (\Omega Y)) \cong H_{n-k} ((\Omega X)\wedge (\Omega Y)) $$ and the Künneth formula provides an isomorphism $$ H_{n+k} ((\Omega X)\wedge (\Omega Y)) \cong H_n((\Omega X) \otimes H_k(\Omega Y)\, . $$ Another application of the Hurewicz theorem shows that $$ H_n(\Omega X) \otimes H_k(\Omega Y) \cong \pi_{n+1}(X) \otimes \pi_{k+1}(Y)\, . $$ Putting this all together, we obtain an isomorphism $$ \pi_{n+k+1} (X\vee Y) \cong \pi_{n-k+1} (X) \oplus\pi_{n-k+1} (Y) \oplus \, \, \pi_{n+1}(X) \otimes \pi_{k+1}(Y)\, . $$\
It remains describe the generalized Whitehead product $\Sigma (\Omega X) \wedge (\Omega Y) \to X\vee Y $. Taking the adjoint, we seek a map $$ (\Omega X) \wedge (\Omega Y) \to \Omega(X\vee Y) $$ Now, there are evident inclusions $ \Omega X \to \Omega(X\vee Y)$ and $\Omega Y \to \Omega(X\vee Y)$. Very roughly, the idea is to map a pair of loops $(\gamma,\omega) \in (\Omega X) \wedge (\Omega Y) $ to the commutator $$ [\gamma,\omega] \in \Omega(X\vee Y) $$ where care is required to make sense of the commutator. I will refrain from writing down the formula here. I believe that the details may be found in Ganea's paper.