Heavy nuclei can be synthetized or entrained in gamma-ray bursts (GRBs) with implications on the high-energy neutrino emission. By means of a Monte-Carlo algorithm, we model nuclear cascades and investigate their impact on the neutrino production considering kinetic dominated jets (in the internal shock model, including a dissipative photosphere) as well as Poynting flux dominated jets (for a jet model invoking internal-collision-induced magnetic reconnection and turbulence, ICMART). We find that the ICMART model allows for efficient nuclear cascades leading to an overall larger neutrino fluence than in the other two jet models. The survival of nuclei and inefficient nuclear cascades lead to an overall reduction of the neutrino fluence up to one order of magnitude. However, if nuclei are disintegrated, the neutrino fluence may be comparable to the one emitted from a jet loaded with protons. Exploring the parameter space of jet properties, we conclude that the composition and the bulk Lorentz factor have significant impact on the efficiency of nuclear cascades as well as the spectral shape of the expected neutrino fluence. On the other hand, the neutrino spectral distribution is less sensitive to the power-law index of the accelerated population of protons or heavier nuclei. For what concerns the diffuse emission of neutrinos from GRBs, we find that the uncertainty due to the jet composition can be at most comparable to the one related to the GRB cosmological rate.