Nuclear magnetic resonance (NMR) is particularly relevant for studies of internuclear spin coupling at zero and ultra-low fields (ZULF), where spin-spin interactions dominate over Zeeman ones. Here we report on ZULF NMR in CdTe. In this semiconductor all magnetic isotopes have spin $I = 1/2$, so that internuclear interactions are never overshadowed by quadrupole effects. Our experiments rely on warm-up spectroscopy, a technique that combines optical pumping, additional cooling via adiabatic demagnetisation, and detection of the oscillating magnetic field-induced warm-up of the nuclear spin system via Hanle effect. We show that NMR spectra exhibit a rich fine structure, consistent with the low abundance of magnetic isotopes in CdTe, their zero quadrupole moments, as well as direct and indirect interactions between them. A model assuming that the electromagnetic radiation is absorbed by nuclear spin clusters composed of up to 5 magnetic isotopes allows us to reproduce the shape of a major part of the measured spectra.