For a Noetherian local ring $(R,\mathfrak m)$, let $K^R$ denote the Koszul complex on a minimal generating set of $\mathfrak m$ (it is well-known that given any two minimal generating set, the Koszul complexes are isomorphic).
My question is: If $x\in \mathfrak m \setminus \mathfrak m^2$ is a non-zero-divisor, then do we have isomorphism of homologies $H_i(K^R)\cong H_i(K^{R/xR})$ for all $i\ge 0$?
Yes. The element $x$ is part of a system of generators of $\mathfrak m$ (lift to $\mathfrak m$ a basis of $\mathfrak m/\mathfrak m^2$ containing the image of $x$). Then use for instance "Bruns-Herzog, 1.6.13.b". More generally, there exists a convergent spectral sequence for a sequence $x_1,...,x_r,...,x_n$
$H_p(x_{r+1},...x_n;H_q(x_1,...,x_r;R)) \implies H_q(x_1,...,x_n;R)$
(this spectral sequence can be seen in the exercises of the last section of "Bourbaki, Algèbre, chap. 10").