  Eliashberg's foundational theory of superconductivity is based on the
application of Migdal's approximation, which states that vertex corrections to
first order electron-phonon scattering are negligible if the ratio between
phonon and electron energy scales is small. The resulting theory incorporates
the first Feynman diagrams for electron and phonon self-energies. However, the
latter is most commonly neglected in numerical analyses. Here we provide an
extensive study of full-bandwidth Eliashberg theory in two and three
dimensions, where we include the full back reaction of electrons onto the
phonon spectrum. We unravel the complex interplay between nesting properties,
Fermi surface density of states, renormalized electron-phonon coupling, phonon
softening, and superconductivity. We propose furthermore a scaling law for the
maximally possible critical temperature $T_c^{\textrm{max}}\propto\lambda
(\Omega ) \sqrt{\Omega_0^2-\Omega^2}$ in 2D and 3D systems, which embodies both
the renormalized electron-phonon coupling strength $\lambda(\Omega)$ and
softened phonon spectrum $\Omega$. Also, we analyze for which electronic
structure properties a maximal $T_c$ enhancement can be achieved.
