We present a chemical analysis of selected HII regions from the PHANGS-MUSE nebular catalogue. Our intent is to empirically re-calibrate strong-line diagnostics of gas-phase metallicity, applicable across a wide range of metallicities within nearby star-forming galaxies. To ensure reliable measurements of auroral line fluxes, we carried out a new spectral fitting procedure whereby only restricted wavelength regions around the emission lines of interest are taken into account: this assures a better fit for the stellar continuum. No prior cuts to nebulae luminosity were applied to limit biases in auroral line detections. Ionic abundances of O^+^, O^2+^, N^+^, S^+^, and S^2+^ were estimated by applying the direct method. We integrated the selected PHANGS-MUSE sample with other existing auroral line catalogues, appropriately re-analysed to obtain a homogeneous dataset. This was used to derive strong-line diagnostic calibrations that span from 12+log(O/H)=7.5 to 8.8. We investigate their dependence on the ionisation parameter and conclude that it is likely the primary cause of the significant scatter observed in these diagnostics. We apply our newly calibrated strong-line diagnostics to the total sample of HII regions from the PHANGS-MUSE nebular catalogue, and we exploit these indirect metallicity estimates to study the radial metallicity gradient within each of the 19 galaxies of the sample. We compare our results with the literature and find good agreement, validating our procedure and findings. With this paper, we release the full catalogue of auroral and nebular line fluxes for the selected HII regions from the PHANGS-MUSE nebular catalogue. This is the first catalogue of direct chemical abundance measurements carried out with PHANGS-MUSE data.