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

We used 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 (MW) stars that has been mapping >6,500 deg^2 of the northern sky with CFHT since 2015. These synthetic magnitudes were used for an absolute recalibration of the deeper Pristine photometry and, combined with broadband Gaia information, synthetic and Pristine CaHK magnitudes were 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 MW ([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 data release 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 spectro-photometric information in Gaia DR3. When available, the much deeper Pristine data greatly enhance the quality of the derived metallicities, in particular at the faint end of the catalogue (G_BP > 16). Combined, both photometric metallicity catalogues include more than two million metal-poor star candidates ([Fe/H]_phot<-1.0) as well as more than 200,000 and ~8,000 very and extremely metal-poor candidates ([Fe/H]_phot<-2.0 and <-3.0, respectively). Finally, we show that these metallicity catalogues can be used efficiently, among other applications to hunt for the most metal-poor stars, and to study how the structure of the MW varies with metallicity.