The recent discoveries of many double neutron star systems and their detection as LIGO-Virgo merger events call for a detailed understanding of their origin. Explosions of ultrastripped stars in binary systems have been shown to play a key role in this context and have also generated interest as a potential explanation for rapidly evolving hydrogen-free transients. Here, we present the first attempt to model such explosions based on binary evolution calculations that follow the mass transfer to the companion to obtain a consistent core-envelope structure as needed for reliable predictions of the supernova transient. We simulate the explosion in 2D and 3D, and confirm the modest explosion energies similar to 10(50) erg and small kick velocities reported earlier in 2D models based on bare carbon-oxygen cores. The spin-up of the neutron star by asymmetric accretion is small in 3D with no indication of spin-kick alignment. Simulations up to shock breakout show the mixing of sizeable amounts of iron group material into the helium envelope. In view of recent ideas for a mixing-length treatment (MLT) of Rayleigh-Taylor instabilities in supernovae, we perform a detailed analysis of the mixing, which reveals evidence for buoyancy-drag balance, but otherwise does not support the MLT approximation. The mixing may have implications for the spectroscopic signatures of ultrastripped supernovae that need to be investigated in the future. Our stellar evolution calculation also predicts presupernova mass-loss due to an off-centre silicon deflagration flash, which suggests that supernovae from extremely stripped cores may show signs of interactions with circumstellar material.