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Characteristic properties of secondary electrons emitted from irradiated two-dimensional materials arise from multi-length and time-scale relaxation processes that connect the initial non-equilibrium excited electron distribution with their eventual emission. To understand these processes, which are critical for using secondary electrons as high-resolution thermalization probes, we combine first-principles real-time electron dynamics with modern experiments. Our data for cold and hot proton-irradiated graphene shows signatures of kinetic and potential emission and generally good agreement for electron yields between experiment and theory. The duration of the emission pulse is about 1.5 femtoseconds, indicating high time resolution when used as a probe. Our newly developed method to predict kinetic energy spectra shows good agreement with electron and ion irradiation experiments and prior models. We find that lattice temperature significantly increases secondary electron emission, whereas electron temperature has a negligible effect.