A theory of spin fluctuations of localized electrons interacting with an optically cooled nuclear spin bath has been developed. Since nuclear spin temperature may stay low enough for a macroscopically long time, the effect of dynamic nuclear polarization on electron spin dynamics can be directly revealed by means of spin noise spectroscopy. We have shown that in the case of weak fields/relatively high nuclear spin temperature, a small degree of nuclear spin polarization affects the electron spin fluctuations in the same way as an additional external magnetic field. By contrast, the high degree of nuclear polarization realized in relatively strong magnetic field and at low nuclear spin temperature leads to a suppression of hyperfine field fluctuations and to a dramatic narrowing of the precession-induced peak in the spin noise spectrum. The experimental possibilities of nuclear spin system investigation by means of spin noise spectroscopy are discussed.