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, 
note 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.