jmc

Taking inspiration from $4^4\mid 10^{10}$ we are inspired to take $n$ to be $p^2$, the lowest prime not dividing $210$, or $11⟹n=121$. Now, there are $242$ factors of $11$, so $11^{242}\mid m^m$, and then $m=11k$ for $k\ge 22$. Now, $\gcd (m+n,210)=\gcd (11+k,210)=1$. Noting $k=26$ is the minimal that satisfies this, we get $(n,m)=(121,286)$. Thus, it is easy to verify this is minimal and we get $\boxed{407}$.