We investigate the phonon-induced relaxation dynamics in the Fe7 magnetic molecule, which is made of two Fe3+ triangles bridged together by a central Fe3+ ion. The competition between different antiferromagnetic exchange interactions leads to a low-spin ground state multiplet with a complex pattern of low-lying excited levels. We theoretically investigate the decay of the time correlation function of molecular observables, such as the cluster magnetization, due to the spin-phonon interaction. We find that more than one time contributes to the decay of the molecular magnetization. The relaxation dynamics is probed by measurements of the nuclear spin-lattice relaxation rate 1/T1. The interpretation of these measurements allows the determination of the magnetoelastic coupling strength and to set the scale factor of the relaxation dynamics time scales. In our theoretical interpretation of 1/T1 data we also take into account the wipeout effect at low temperatures.