High precision single-differential $W^\pm$-boson production cross-sections as a function of electron or muon transverse momentum $p_\mathrm{T}$ or their pseudorapity $\eta$, as well as double-differential cross-sections as functions of these variables, are measured in proton-proton collisions at centre-of-mass energies $\sqrt{s}=5.02$ TeV and 13 TeV. The $W$-boson charge asymmetry as a function of lepton $\eta$ is also measured. The data, collected in dedicated runs at reduced instantaneous luminosity with the ATLAS detector at the Large Hadron Collider, correspond to integrated luminosities of 255 pb$^{-1}$ at 5.02 TeV and 338 pb$^{-1}$ at 13 TeV. The measurements are in agreement with Standard-Model predictions calculated at next-to-next-to-leading-order in the strong coupling constant $\alpha_s$ including transverse-momentum resummation at next-to-next-to-leading logarithmic accuracy using several parton distribution functions. The impact of the measured differential cross-sections as a function of lepton $\eta$ on the determination of these functions is studied using a profiling technique.