The total and differential Higgs boson production cross-sections are measured through a combined statistical analysis of the H → ZZ ^* → 4ℓ and H → γγ decay channels. The results are based on a dataset of 139 fb ⁻¹ of proton–proton collisions at a centre-of-mass energy of 13 TeV, recorded by the ATLAS detector at the Large Hadron Collider. The measured total Higgs boson production cross-section is 55.5−3.8+4.0 pb, consistent with the Standard Model prediction of 55.6 ± 2.5 pb. All results from the two decay channels are compatible with each other, and their combination agrees with the Standard Model predictions. A combined statistical interpretation of the measured fiducial cross-sections as a function of the Higgs boson transverse momentum is performed in order to probe the Yukawa couplings to the bottom and charm quarks. A similar interpretation is performed by including also the constraints from the measurements of Higgs boson production in association with a W or Z boson in the H → bb¯ and cc¯ decay channels. [Figure not available: see fulltext.]