Here we predict, using a semimolecular theory, the Stokes shift dynamics of a dipolar solute in binary mixtures of an ionic liquid (IL), 1-butyl-3-methylimidazolium tetrafluoroborate ([Bmim][BF4]), with water (H2O) and acetonitrile (CH3CN), and compare with the experimental results. The latter are from the recent measurements that combined broad-band fluorescence up-conversion (FLUPS) with time-correlated single photon counting (TCSPC) techniques and used coumarin 153 (C153) as a solute probe. Nine different compositions of ([Bmim][BF4] + H2O) and ([Bmim][BF4] + CH3CN) binary mixtures are considered for the extensive comparison between theory and experiments. Two separate model calculations have been performed using the available experimental frequency dependent dielectric function, epsilon(omega). These calculations semi-quantitatively reproduce the experimentally observed (i) IL mole fraction dependence of dynamic Stokes shifts in these mixtures, (ii) composition dependence of average fast, slow, and solvation times, (iii) viscosity dependence of slow times, and (iv) the nonlinear dependence of average solvation times on experimental inverse conductivity. Variations of the calculated dynamics on water dipole moment values (gas phase or liquid phase) and sensitivity to different measurements of epsilon(omega) for ([Bmim][BF4] + H2O) mixtures are examined. In addition, the importance of the missing contribution to experimental epsilon(omega) from high frequency collective solvent intermolecular modes for generating the experimentally observed sub-picosecond solvation response in these (IL + polar solvent) binary mixtures has been explored.