We investigate evolutionary pathways leading to neutron star (NS) formation through the collapse of oxygen–neon white dwarf (ONe WD) stars in interacting binaries. We consider (1) non-dynamical mass transfer where an ONe WD approaches the Chandrasekhar mass leading to accretion-induced collapse (AIC) and (2) dynamical time-scale merger-induced collapse (MIC) between an ONe WD and another WD. We present rates, delay times, and progenitor properties for two different treatments of common envelope evolution. We show that AIC NSs are formed via many different channels and the most dominant channel depends on the adopted common envelope physics. Most AIC and MIC NSs are born shortly after star formation, though some have delay times >10 Gyr. The shortest delay time (25–50 Myr) AIC NSs have stripped-envelope, compact, helium-burning star donors, though many prompt AIC NSs form via wind accretion from an asymptotic giant branch star. The longest delay time AIC NSs, which may be observed as young millisecond pulsars among globular clusters, have a red giant or main-sequence donor at the time of NS formation and will eventually evolve into NS  + helium WD binaries. We discuss AIC and MIC binaries as potential gravitational wave sources for Laser Interferometer Space Antenna (LISA). NSs created via AIC undergo a low-mass X-ray binary phase, offering an electromagnetic counterpart for those shortest orbital period sources that LISA could identify. The formation of NSs from interacting WDs in binaries is likely to be a key mechanism for the production of LIGO/Virgo gravitational wave sources (NS–NS and BH–NS mergers) in globular clusters.