The electronic and vibrational properties and electron-phonon couplings of one-dimensional materials will be key to many prospective applications in nanotechnology. Dimensionality strongly affects these properties and has to be correctly accounted for in first-principles calculations. Here we develop and implement a formulation of density functional and density functional perturbation theory that is tailored for one-dimensional systems. A key ingredient is the inclusion of a Coulomb cutoff, a reciprocal-space technique designed to correct for the spurious interactions between periodic images in periodic-boundary conditions. This restores the proper one-dimensional open-boundary conditions, letting the true response of the isolated one-dimensional system emerge. In addition to total energies, forces and stress tensors, phonons and electron-phonon interactions are also properly accounted for. We demonstrate the relevance of the present method on a portfolio of realistic systems: BN atomic chains, BN armchair nanotubes, and GaAs nanowires. Notably, we highlight the critical role of the Coulomb cutoff by studying previously inaccessible polar-optical phonons and Fröhlich electron-phonon couplings. We also develop and apply analytical models to support the physical insights derived from the calculations and we discuss their consequences on electronic lifetimes. The present study unlocks the possibility to accurately simulate the linear-response properties of one-dimensional systems, sheds light on the transition between dimensionalities, and paves the way for further studies in several fields, including charge transport, optical coupling, and polaritronics.