We have the identity $$\gcd\left(\frac{a^n-b^n}{a-b},a-b\right)=\gcd(n\gcd(a,b)^{n-1},a-b).$$ (see here)
This appears to be a quite useful result with various applications. I wonder whether there is a similar identity for
$$\gcd\left(\frac{a^n-b^n}{a-b},a^m-b^m\right),$$
preferably telling what this $\gcd$ is in terms of $n,m,a-b$ or $a^{\gcd(m,n)}-b^{\gcd(m,n)}$ or so.