  We study a multi-orbital Hubbard--Su-Schrieffer-Heeger model for the
one-dimensional (1D) corner-shared cuprates in the adiabatic and nonadiabatic
limits using the exact diagonalization and determinant quantum Monte Carlo
methods. Our results demonstrate that lattice dimerization can be achieved only
over a narrow range of couplings slightly below a critical coupling $g_c$ at
half-filling. Beyond this critical coupling, the sign of the effective hopping
changes, and the lattice becomes unstable. We also examine the model's
temperature-dependent uniform magnetic susceptibility and the dynamical
magnetic susceptibility and compare them to the results of an effective
spin-$1/2$ Heisenberg model. In doing so, we numerically demonstrate that the
lattice fluctuation induced by the $e$-ph interaction suppresses the effective
superexchange interaction. Our results elucidate the effect of bond-stretching
phonons in the parent cuprate compounds in general and are particularly
relevant to 1D cuprates, where strong $e$-ph interactions have recently been
inferred.
