It is being increasingly realized that nucleosome organization on DNA crucially regulates DNA-protein interactions and the resulting gene expression. While the spatial character of the nucleosome positioning on DNA has been experimentally and theoretically studied extensively, the temporal character is poorly understood. Accounting for ATPase activity and DNA-sequence effects on nucleosome kinetics, we develop a theoretical method to estimate the time of continuous exposure of binding sites of non-histone proteins (e.g. transcription factors and TATA binding proteins) along any genome. Applying the method to Saccharomyces cerevisiae, we show that the exposure timescales are determined by cooperative dynamics of multiple nucleosomes, and their behavior is often different from expectations based on static nucleosome occupancy. Examining exposure times in the promoters of GAL1 and PHO5, we show that our theoretical predictions are consistent with known experiments. We apply our method genome-wide and discover huge gene-to-gene variability of mean exposure times of TATA boxes and patches adjacent to TSS (+1 nucleosome region); the resulting timescale distributions have non-exponential tails.