We perform hydrodynamic simulations of mass transfer in binaries that contain a white dwarf and a neutron star (WD–NS binaries), and measure the specific angular momentum of material lost from the binary in disc winds. By incorporating our results within a long-term evolution model, we measure the long-term stability of mass transfer in these binaries. We find that only binaries containing helium white dwarfs (WDs) with masses less than a critical mass of M_{WD, crit} = 0.2 M_⊙ undergo stable mass transfer and evolve into ultracompact X-ray binaries. Systems with higher mass WDs experience unstable mass transfer, which leads to tidal disruption of the WD. Our low critical mass compared to the standard jet-only model of mass-loss arises from the efficient removal of angular momentum in the mechanical disc winds, which develop at highly super-Eddington mass-transfer rates. We find that the eccentricities expected for WD–NS binaries when they come into contact do not affect the loss of angular momentum, and can only affect the long-term evolution if they change on shorter time-scales than the mass-transfer rate. Our results are broadly consistent with the observed numbers of both ultracompact X-ray binaries and radio pulsars with WD companions. The observed calcium-rich gap transients are consistent with the merger rate of unstable systems with higher mass WDs.