Abstract: The recent observation of coherent elastic neutrino-nucleus scattering (CEvNS) with neutrinos from pion decay at rest (N-DAR) sources by the COHERENT Collaboration has raised interest in this process in the search for new physics. Unfortunately, current uncertainties in the determination of nuclear parameters relevant to those processes can hide new physics effects. This is not the case for processes involving lower-energy neutrino sources such as nuclear reactors. Note, however, that a CEvNS measurement with reactor neutrinos depends largely on a (still-missing) precise determination of the quenching factor at very low energies, making its observation more challenging. In the upcoming years, once this signal is confirmed, a combined analysis of N-DAR and reactor CEvNS experiments will be very useful to probe particle and nuclear physics, with a reduced dependence on nuclear uncertainties. In this work, we explore this idea by simultaneously testing the sensitivity of current and future CEvNS experiments to neutrino nonstandard interactions (NSIs) and the neutron root mean square (rms) radius, considering different neutrino sources as well as several detection materials. We show how the interplay between future reactor and accelerator CEvNS experiments can help to get robust constraints on the neutron rms and to break degeneracies between the NSI parameters. Our forecast could be used as a guide to optimize the experimental sensitivity to the parameters under study.