Ultra-high-energy neutrinos and cosmic rays are excellent probes of astroparticle physics phenomena. For astroparticle physics analyses, robust and accurate reconstruction of signal parameters like arrival direction and energy is essential. Current reconstruction methods ignore bin-to-bin noise correlations, which limits reconstruction resolution and so far has prevented calculations of event-by-event uncertainties. In this work, we present a likelihood description of neutrino or cosmic-ray signals in a radio detector with correlated noise, as present in all neutrino and cosmic-ray radio detectors. We demonstrate with a toy-model reconstruction that signal parameters such as energy and direction, including event-by-event uncertainties with correct coverage, can be obtained. Additionally, by correctly accounting for correlations, the likelihood description constrains the best-fit parameters better than alternative methods and thus improves experimental reconstruction capabilities.