$\chi(G)+\chi(G')\leq n+1$

There is a straightforward proof by induction on the number of vertices of $G$. I leave the induction base to you.

For the induction step, let $G$ be a graph on $n+1$ vertices, $v$ a vertex of $G$. So (by the induction hypothesis) we have a $k$-coloring of $G-v$ and an $l$-coloring of $G'-v$, such that $k+l=n+1$.

Now if $d_G(v)<k$ we can extend the coloring on $G-v$ to a proper $k$-coloring of $G$ and use a new color in $G'$ for a total of at most $n+2$ colors. Otherwise $d_G(v)\geq k$, so $d_{G'}(v)\leq n-k=l-1<l$, so we can extend the coloring on $G'-v$ to a proper $l$-coloring of $G'$ and use a new color on $G$.