  The large variations of T$_c$ across the cuprate families is one of the major
unsolved puzzles in condensed matter physics, and is poorly understood.
Although there appears to be a great deal of universality in the cuprates,
several orders of magnitude changes in T$_c$ can be achieved through changes in
the chemical composition and structure of the unit cell. In this paper we
formulate a systematic examination of the variations in electron-phonon
coupling to oxygen phonons in the cuprates, incorporating a number of effects
arising from several aspects of chemical composition and doping across cuprate
families. It is argued that the electron-phonon coupling is a very sensitive
probe of the material-dependent variations of chemical structure, affecting the
orbital character of the band crossing the Fermi level, the strength of local
electric fields arising from structural-induced symmetry breaking, doping
dependent changes in the underlying band structure, and ionicity of the crystal
governing the ability of the material to screen $c$-axis perturbations. Using
electrostatic Ewald calculations and known experimental structural data, we
establish a connection between the material's maximal T$_c$ at optimal doping
and the strength of coupling to $c$-axis modes. We demonstrate that materials
with the largest coupling to the out-of-phase bond-buckling (``$B_{1g}$")
oxygen phonon branch also have the largest T$_c$'s. In light of this
observation we present model T$_c$ calculations using a two-well model where
phonons work in conjunction with a dominant pairing interaction, presumably due
to spin fluctuations, indicating how phonons can generate sizeable enhancements
to T$_c$ despite the relatively small coupling strengths. Combined, these
results can provide a natural framework for understanding the doping and
material dependence of T$_c$ across the cuprates.
