We report the first synthesis of the intermetallic compound ${\mathrm{Ni}}_{5}{\mathrm{Al}}_{3}$ in the nanocrystalline form. At the nanometer length scale, a stable ${\mathrm{Ni}}_{5}{\mathrm{Al}}_{3}$ compound (core) does not form without a NiO shell. Detailed structural and microstructural characterization as well as the compositional analysis indicate the presence of ${\mathrm{Ni}}_{5}{\mathrm{Al}}_{3}/\mathrm{NiO}$ core/shell nanoparticles and log-normal crystallite size distribution with mean size $d\phantom{\rule{4pt}{0ex}}\ensuremath{\simeq}$ 6 nm. ``Zero-field'' (dc magnetic field, $H=0$) linear (${\ensuremath{\chi}}_{1}$) and nonlinear (${\ensuremath{\chi}}_{n}$ with $n=2,3,4,5$) ac-magnetic susceptibilities have been measured as functions of temperature at various ac driving field amplitudes over three decades of frequency on the nanocrystalline samples ${\mathrm{S}}_{1}$ and ${\mathrm{S}}_{2}$ of composition ${\mathrm{Ni}}_{5+x}{\mathrm{Al}}_{3\ensuremath{-}x}/\mathrm{NiO}$ ($x=0.014$) and ${\mathrm{Ni}}_{5\ensuremath{-}x}{\mathrm{Al}}_{3+x}/\mathrm{NiO}$ ($x=0.038$), respectively. ${\ensuremath{\chi}}_{1}(T)$ and ${\ensuremath{\chi}}_{n}(T)$ are shown to provide conclusive evidence for the existence of two spin glass (SG) thermodynamic phase transitions: one at ${T}_{i}(H=0)\ensuremath{\simeq}138$ K and the other at a lower temperature ${T}_{p}(H=0)\ensuremath{\simeq}108$ K in both ${\mathrm{S}}_{1}$ and ${\mathrm{S}}_{2}$. Linear ac-susceptibility in superposed dc magnetic fields demonstrates that the thermodynamic nature of these transitions is preserved in finite fields and the $H\ensuremath{-}T$ phase diagrams for the samples ${\mathrm{S}}_{1}$ and ${\mathrm{S}}_{2}$ conform very well only with the $H\ensuremath{-}T$ phase diagram predicted by the chirality-driven spin glass (SG) ordering model for a three-dimensional nearest-neighbor Heisenberg SG system with weak random anisotropy. A detailed comparison between theory and experiment permits us to unambiguously identify various ``zero-field'' and ``in-field'' SG phase transitions as (i) the simultaneous paramagnetic (PM): chiral-spin glass (CG) and PM-SG phase transitions at ${T}_{i}(H)$ and (ii) the replica symmetry-breaking SG transition at ${T}_{p}(H)$.