Nanowires made of chalcogenide alloys are of interest for use in phase-change nonvolatile memories. For this application, insights into the thermal properties of such nanowires and, in particular, into the crystallization kinetics at the atomic level are crucial. Toward this end, we have performed large-scale atomistic simulations of ultrathin nanowires (9 nm in diameter) of the prototypical phase change compound GeTe. We made use of an interatomic potential generated by the neural network fitting of a large ab initio database to compute the thermal properties of the nanowires. By melting a portion of a nanowire, we investigated the velocity of recrystallization as a function of temperature. The simulations show that the melting temperature of the nanowire is about 100 K below the melting temperature of the bulk, which yields a reduction by about a factor of 2 of the maximum crystallization speed. Further, analysis of the structural properties of the amorphous phase of the nanowire suggests a possible origin of the reduction of the resistance drift observed experimentally in nanowires with respect to the bulk.