The Pristine survey -- XXIII. Data Release 1 and an all-sky metallicity catalogue based on Gaia DR3 BP/RP spectro-photometry

We use the spectro-photometric information of ~219 million stars from Gaia's DR3 to calculate synthetic, narrow-band, metallicity-sensitive CaHK magnitudes that mimic the observations of the Pristine survey, a survey of photometric metallicities of Milky Way stars that has been mapping more than 6,500 deg^2 of the northern sky with the CFHT since 2015. These synthetic magnitudes are used for an absolute re-calibration of the deeper Pristine photometry and, combined with broadband Gaia information, synthetic and Pristine CaHK magnitudes are used to estimate photometric metallicities over the whole sky. The resulting metallicity catalogue is accurate down to [Fe/H]~-3.5 and is particularly suited for the exploration of the metal-poor Milky Way ([Fe/H]<-1.0). We make available here the catalogue of synthetic CaHK_syn magnitudes for all stars with BP/RP information in Gaia DR3, as well as an associated catalogue of more than ~30 million photometric metallicities for high S/N FGK stars. This paper further provides the first public DR of the Pristine catalogue in the form of higher quality recalibrated Pristine CaHK magnitudes and photometric metallicities for all stars in common with the BP/RP information in Gaia DR3. We demonstrate that, when available, the much deeper Pristine data greatly enhances the quality of the derived metallicities, in particular at the faint end of the catalogue (G_BP>16). Combined, both catalogues include more than 2 million metal-poor star candidates as well as more than 200,000 and ~8,000 very and extremely metal-poor candidates. Finally, we show that these metallicity catalogues can be used efficiently, among other applications, for Galactic archaeology, to hunt for the most metal-poor stars, and to study how the structure of the Milky Way varies with metallicity, from the flat distribution of disk stars to the spheroid-shaped metal-poor halo. (Shortened)