We have selectively studied the component dynamics in a nanocomposite where 25 wt % of PMMA [poly(methyl methacrylate)] soft nanoparticles (SNPs) are dispersed in PEO [poly(ethylene oxide)] by means of quasi-elastic neutron scattering (QENS) experiments on partially deuterated samples. We have covered a time range from subpico to nanosecond regime and a momentum transfer range 0.5 ≤ Q ≤ 1.8 Å-1 by combining three different spectrometers. Complementary diffraction measurements with polarization analysis have facilitated the data analysis, by providing the coherent and incoherent contributions to the scattered intensities. Regarding the SNPs, the α-methyl group dynamics of PMMA - to which the QENS experiments are most sensitive in the temperature range investigated - turn to be faster than in bulk PMMA. This could be due to the plasticization effect induced by the fast PEO chains. In fact, calorimetric measurements show the coexistence of two glass-transition temperatures in the system, associated with each of the components, but modified with respect to those in the neat materials. The QENS results on the PEO component for large length scales reveal Rouse-like dynamics slowed down by the presence of the SNPs with respect to that in the bulk. With decreasing temperature indications for distributed chain mobilities are found, probably due to the enhancement of the concentration fluctuations. At local scales, deviations from Rouse-like dynamics occur, that could be attributed to an extra-friction related to the local potentials, and also to non-Gaussian effects arising from the discrete character of the elementary processes underlying the subdiffusive dynamics in the polymer. The deviations take place in a very similar way as in bulk PEO.