We present new cooling models for carbon-oxygen white dwarfs (WDs) with both H- and He-atmospheres, covering the whole relevant mass range, to extend our updated BASTI (a Bag of Stellar Tracks and Isochrones) stellar evolution archive. They have been computed using core chemical stratifications obtained from new progenitor calculations, adopting a semi-empirical initial-final mass relation. The physics inputs have been updated compared to our previous BASTI calculations: 22Ne diffusion in the core is now included, together with an updated CO phase diagram, and updated electron conduction opacities. We have calculated models with various different neon abundances in the core, suitable to study WDs in populations with metallicities ranging from supersolar to metal poor, and have performed various tests/comparisons of the chemical stratification and cooling times of our models. Two complete sets of calculations are provided, for two different choices of the electron conduction opacities, to reflect the current uncertainty in the evaluation of the electron thermal conductivity in the transition regime between moderate and strong degeneracy, crucial for the H- and He-envelopes. We have also made a first, preliminary estimate of the effect - that turns out to be generally small - of Fe sedimentation on the cooling times of WD models, following recent calculations of the phase diagrams of carbon-oxygen-iron mixtures. We make publicly available the evolutionary tracks from both sets of calculations, including cooling times and magnitudes in the Johnson-Cousins, Sloan, Pan-STARSS, GALEX, Gaia-DR2, Gaia-eDR3, HST-ACS, HST-WFC3, and JWST photometric systems.