We present the measurements of the proton spin-lattice relaxation time T1 of liquid crystal 4-n-octyl-4'-cyanobiphenyl (8CB) confined into randomly oriented approximately 15 nm pores of untreated porous glass. In the low kilohertz range the spin-lattice relaxation rate in the nanoconfined 8CB is about ten times larger than in the bulk. We show that the increase is mainly due to molecular reorientations mediated by translational displacements (RMTD). In the paranematic phase the power law describing the RMTD dispersion, (T1(-1))RMTD proportional, omega(-p), is well characterized by the exponent p=0.5+/-0.06 and suggests an equipartition of diffusion modes with different wavelengths. The largest distance related to the decay of the orientational correlation function is about twice the diameter of the cavity. The situation is different in the nematic phase, where the orientational correlation is eventually lost at approximately 60 nm in the direction along the pore, a distance corresponding roughly to the length of a pore segment in the glassy matrix. The exponent p is between 0.65 and 0.9, depending on the temperature, which implies that in the nematic phase long wavelength modes are relatively more important--a consequence of the uniform director field along the pore. These observations are in agreement with the model of mutually independent pores with nematic director parallel to the pore axis in each segment. We point out that in strongly confined liquid crystals the proton NMR relaxometry does not provide the evidence of director fluctuations correlated over micrometer distances as was suggested earlier. The local translational diffusion of molecules within the cavities is found about as fast as in bulk.