When do the first neurons appear in humans?
Apparently, olfactory axons and GnRH producing neurons are among the first neurons that migrate from neural crest at around 39th day of gestation in humans (Cassoni et al., 2016). 


  In rodents, early GnRH neurons migrate together with a heterogeneous
  coalescence of placode-derived and neural crest-derived migratory
  cells (Forni et al., 2011) and olfactory axons, collectively called
  the ‘migratory mass’ (MM) (Miller et al., 2010; Valverde et al.,
  1992). This cell migration precedes the targeting of olfactory sensory
  axons to the developing olfactory bulb (OB). The existence of a
  similar MM in the human embryo has not yet been described. We thus
  immunolabeled consecutive sagittal sections of a CS 16 embryo&dagger; for GnRH
  and doublecortin (DCX) (Fig. 1D,E,G), a marker of immature migratory
  neurons (Gleeson et al., 1999). We identified a very small number (50
  in total) of immature GnRH-expressing cells in the nasal mesenchyme,
  in the medial portion of the olfactory placode (Fig. 1C-E), adjacent
  to the basal lamina of the VNO, showing that the acquisition of cell
  identity occurs outside the VNO between GW 5 and 6. At this stage, we
  observed a mixed mass of immature GnRH neurons expressing DCX (Fig.
  1F,G) or βIII-tubulin (Fig. 1H-J), migrating across the nasal
  mesenchyme towards the telencephalon. As in rodents (Miller et al.,
  2010), GnRH neurons in humans only represented a small proportion of
  the MM (Fig. 1G). Furthermore, these pioneer neurons of the MM
  expressed the delta/notch-like EGF repeat containing (DNER) (Fig.
  1H-J), a transmembrane protein specifically localized in the dendrites
  and cell bodies of postmitotic neurons.
  
  &dagger;Carnegie stage (CS) 16 = ∼39th day of gestation