Amplitudes derived from scattering data on elementary targets are basic inputs to neutrino-nucleus cross section predictions. A prominent example is the isovector axial nucleon form factor, FA(q2), which controls charged current signal processes at accelerator-based neutrino oscillation experiments. Previous extractions of FA from neutrino-deuteron scattering data rely on a dipole shape assumption that introduces an unquantified error. A new analysis of world data for neutrino-deuteron scattering is performed using a model-independent, and systematically improvable, representation of FA. A complete error budget for the nucleon isovector axial radius leads to rA2=0.46(22) fm2, with a much larger uncertainty than determined in the original analyses. The quasielastic neutrino-neutron cross section is determined as σ(νμn→μ-p)|Eν=1 GeV=10.1(0.9)×10-39 cm2. The propagation of nucleon-level constraints and uncertainties to nuclear cross sections is illustrated using MINERvA data and the GENIE event generator. These techniques can be readily extended to other amplitudes and processes.