The magnetic molecular cluster $[{\mathrm{Fe}}_{8}({\mathrm{N}}_{3}{\mathrm{C}}_{6}{\mathrm{H}}_{15}{)}_{6}{\mathrm{O}}_{2}(\mathrm{OH}{)}_{12}{]}^{8+}[{\mathrm{Br}}_{8}\ensuremath{\cdot}9{\mathrm{H}}_{2}\mathrm{O}{]}^{8\ensuremath{-}},$ in short Fe8, has been investigated at low temperature by ${}^{1}\mathrm{H}$-NMR and relaxation measurements. Some measurements of ${}^{2}\mathrm{D}$-NMR in partially deuterated Fe8 clusters will also be reported. Upon decreasing temperature the NMR spectra display a very broad and structured signal which is the result of the internal local fields at the proton sites due to the local moments of the Fe(III) ions in the total $S=10$ magnetic ground state. The proton and deuteron NMR spectra have been analyzed and the different resonance peaks have been attributed to the different proton groups in the molecule. The simulation of the spectra by using a dipolar hyperfine field and the accepted model for the orientation of the Fe(III) local moments do not agree with the experiments even when the magnitude of the local Fe(III) moments is allowed to vary. It is concluded that a positive contact hyperfine interaction of the same order of magnitude as the dipolar interaction is present for all proton sites except the water molecules. The temperature and magnetic field dependence of the nuclear spin-lattice relaxation rate is ascribed to the fluctuations of the local Fe(III) moments, which follow rigidly the fluctuations of the total ground state magnetization of the nanomagnet. By using a simple model already utilized for the Mn12 cluster, we derive the value of the spin phonon coupling constant which determines the lifetime broadening of the different magnetic quantum number m substates of the $S=10$ ground state. It is shown that the lifetime broadening decreases rapidly on lowering the temperature. When the lifetime becomes longer than the reciprocal of the frequency shift of the proton lines a structure emerges in the NMR spectrum reflecting the ``frozen'' local moment configuration.