We present a detailed, 3D hydrodynamics study of the neutrino-driven winds that emerge from the remnant of a NS merger. Our simulations are performed with the Newtonian, Eulerian code FISH, augmented by a detailed, spectral neutrino leakage scheme that accounts for heating due to neutrino absorption in optically thin conditions. Consistent with the 2D study of Dessart et al. (2009), we find that a strong baryonic wind is blown out along the original binary rotation axis within $100$ ms after the merger. We compute a lower limit on the expelled mass of $3.5 \times 10^{-3} M_{\odot}$, large enough to be relevant for heavy element nucleosynthesis. The physical properties vary significantly between different wind regions. For example, due to stronger neutrino irradiation, the polar regions show substantially larger $Y_e$ than those at lower latitudes. This has its bearings on the nucleosynthesis: the polar ejecta produce interesting r-process contributions from $A\sim 80$ to about 130, while the more neutron-rich, lower-latitude parts produce also elements up to the third r-process peak near $A\sim 195$. We also calculate the properties of electromagnetic transients that are powered by the radioactivity in the wind, in addition to the macronova transient that stems from the dynamic ejecta. The high-latitude (polar) regions produce UV/optical transients reaching luminosities up to $10^{41} {\rm erg \, s^{-1}}$, which peak around 1 day in optical and 0.3 days in bolometric luminosity. The lower-latitude regions, due to their contamination with high-opacity heavy elements, produce dimmer and more red signals, peaking after $\sim 2$ days in optical and infrared. Our numerical experiments indicate that it will be difficult to infer the collapse time-scale of the HMNS to a BH based on the wind electromagnetic transient, at least for collapse time-scales larger than the wind production time-scale.