A theoretical study of the nuclear spin-lattice relaxation in cubic ferromagnetic insulators at ultralow temperatures is presented. Calculations are performed for nuclei which belong to the magnetic atoms, considering only the direct processes. Three mechanisms are considered: the relaxation to mixed magnon-phonon modes, indirect nuclear-spin interaction modulated by lattice vibrations, and nuclear quadrupole energy modulated by lattice vibrations. The first two mechanisms lead to a relaxation time T1 which depends on both the temperature T and the external field H0, with T1~H02T for H0>> magnetization and the anisotropy field. For a field comparable to the magnetization or the anisotropy field, the relaxation time is proportional to H0n, n being larger or smaller than 2 depending on the shape of the single-crystal sample. The last mechanism does not lead to a field-dependent T1. Comparison with experiments performed on powdered EuS is also presented.