Gravitational-wave (GW) observations of neutron star-black hole (NSBH) mergers are sensitive to the nuclear equation of state (EOS). We present a new methodology for EOS inference with nonparametric Gaussian process priors, enabling direct constraints on the pressure at specific densities and the length-scale of correlations on the EOS. Using realistic simulations of NSBH mergers, incorporating both GW and electromagnetic selection to ensure sample purity, we find that a GW detector network operating at O5 sensitivities will constrain the radius of a 1.4M⊙ NS and the maximum NS mass with 1.6% and 13% precision, respectively. With the same sample, the projected constraint on the length-scale of correlations in the EOS is ≥3.2 MeV fm-3. These results demonstrate strong potential for insights into the nuclear EOS from NSBH systems, provided they are robustly identified.