A comprehensive study of the magnetization dynamics of oxygen-stabilized, tetragonal $\mathrm{Ni}\phantom{\rule{0.3em}{0ex}}(\mathrm{OS}\text{\ensuremath{-}}\mathrm{Ni})$ nanoparticles $(20\phantom{\rule{0.3em}{0ex}}\mathrm{nm})$ prepared by the borohydride reduction method is reported. Both as-prepared and air annealed (at $573\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and $773\phantom{\rule{0.3em}{0ex}}\mathrm{K}$) samples have been studied using ac susceptibility, aging experiments, and field cooled (FC) and zero field cooled (ZFC) magnetization measurements as a function of temperature. The OS structure, which is a deviation from the usual face-centered-cubic (fcc) structure of Ni, arises due to the presence of interstitial oxygen atoms in the unit cell of Ni and causes it to exhibit anomalous magnetic behavior such as a very large magnetization enhancement at low temperatures. Two low-temperature magnetic transitions in close succession, manifested in the form of a sharp peak at $20\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and a small hump at $\ensuremath{\sim}12\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, are observed in the ZFC curve and ac susceptibility plots of the as-prepared sample. The probable nature of these transitions has been explained on the basis of a model which associates the peak at $20\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ with the occurrence of a $\mathrm{PM}\phantom{\rule{0.3em}{0ex}}(\mathrm{paramagnetic})\ensuremath{\rightarrow}\mathrm{FM}\phantom{\rule{0.3em}{0ex}}(\mathrm{ferromagnetic})$ transition of the oxygen-stabilized phase. Most of these newly formed FM particles block as soon as they gain internal order, yielding a strong irreversibility between the FC and ZFC branches. Some of the macromoments formed at $20\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, however, remain unstable down to $12\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, the temperature at which they block cooperatively as shown by a critical dynamics analysis, yielding critical exponent $z\ensuremath{\nu}=9.84\ifmmode\pm\else\textpm\fi{}0.48$ and a relaxation prefactor of ${\ensuremath{\tau}}_{0}={10}^{\ensuremath{-}7}\phantom{\rule{0.3em}{0ex}}\mathrm{s}$. Aging experiments at $10\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ for three different wait times ${t}_{\mathrm{w}}$ show wait time dependency, substantiating unequivocally the cooperative, chaotic nature of the sample magnetic dynamics. The low-temperature magnetic features displayed by the $573\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ annealed sample closely resemble those of the as-prepared one, though distinctly different features are observed in the sample annealed at $773\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. These have been explained coherently taking into account the structural changes produced upon annealing.