The recent detection of super-solar carbon-to-oxygen and nitrogen-to-oxygen abundance ratios in a bunch of metal-poor galaxies at high redshift by the James Webb Space Telescope has sparked renewed interest in exploring the chemical evolution of carbon, nitrogen, and oxygen (the CNO elements) at early times, prompting fresh inquiries into their origins. The main goal of this paper is to shed light onto the early evolution of the main CNO isotopes in our Galaxy and in young distant systems, such as GN-z11 at z=10.6 and GS-z12 at z=12.5. To this aim, we incorporate a stochastic star-formation component into a chemical evolution model calibrated with high quality Milky Way (MW) data, focusing on the contribution of Population III (Pop III) stars to the early chemical enrichment. By comparing the model predictions with CNO abundance measurements from high-resolution spectroscopy of an homogeneous sample of Galactic halo stars, we first demonstrate that the scatter observed in the metallicity range -4.5<=[Fe/H]<=-1.5 can be explained by pre-enrichment from Pop III stars that explode as supernovae (SNe) with different initial masses and energies. Then, by exploiting the chemical evolution model, we provide testable predictions for log(C/N), log(N/O), and log(C/O) vs. log(O/H)+12 in MW-like galaxies observed at different cosmic epochs/redshifts. Finally, by calibrating the chemical evolution model to replicate the observed properties of GN-z11 and GS-z12, we provide an alternative interpretation of their high N/O and C/O abundance ratio, respectively, demonstrating that an anomalously high N or C content can be reproduced through enrichment from faint Pop III SNe. Stochastic chemical enrichment from primordial stars explains both the observed scatter in CNO abundances in MW halo stars and the exceptionally high C/O and N/O ratios in some distant galaxies. These findings emphasize the critical role of Pop III stars in shaping early chemical evolution.