Context. Stellar models and nucleosynthetic yields of primordial to extremely metal-poor (EMP) stars are crucial to interpret the surface abundances of the most metal-poor stars observed and, ultimately, to better understand the earliest stellar populations. In addition, they are key ingredients of Galactic chemical evolution models. Aims. We aim to better characterise the evolution and fates, and determine updated nucleosynthetic yields of intermediate-mass stars between primordial and EMP metallicity (Z = 10−10, 10−8, 10−7, 10−6, and 10−5). We also probed uncertainties in the nucleosynthesis of the oldest intermediate-mass stars, namely those related to the treatment of convection and convective boundaries and those related to wind prescriptions during the asymptotic giant branch (AGB) phase. Methods. We analyse the evolution of models from their main sequence, through the thermally pulsing AGB (TP-AGB), to the latest stages of their evolution, using the Monash-Mount Stromlo stellar evolution code MONSTAR. The results were post-processed with the code MONSOON, which allowed for the determination of the nucleosynthetic yields of 77 species up to 62Ni. By comparing them to similar calculations existing in the literature, we inspected the effects of input physics on the nucleosynthesis of EMP models. Results. From the evolutionary point of view, as reported in former works, we identified proton ingestion episodes (PIEs) in our lowest-mass lowest-metallicity models. Models of Z = 10−10 and Z = 10−8 in a narrow initial mass range around 5 M⊙ experience the cessation of thermal pulses, and their final fates as type-I1/2 supernovae cannot be discarded. However, the initial mass range of models eventually leading to the formation of type-I1/2 and electron-capture supernovae is considerably reduced compared to former works. All the models of initial mass ≳6–7 M⊙ experience a corrosive second dredge-up and, analogously to those experiencing PIEs, undergo significant metal enrichment in their envelopes. The associated increase in their opacities allows them to develop a solar-like TP-AGB or TP-super-AGB, ultimately becoming white dwarfs. Except for those undergoing the cessation of thermal pulses, all of our models show the nucleosynthetic signatures of both efficient third dredge-up and hot-bottom burning, with the activation of the NeNa cycle and the MgAlSi chains. This leads to the creation of vast amounts of CNO, with typical [N/Fe] > 4), and the characteristic abundance signature [N/Fe] > [C/Fe] > [O/Fe]. Our nucleosynthetic yields present dramatic differences with respect to recent results existing in the literature for intermediate-mass models of similar metallicities. The reason for these discrepancies lay in the poorly known input physics related to stellar winds and, above all, the treatment of convection and convective boundaries.