Context. Understanding planet formation is important in the context of the origin of planetary systems in general and of the Solar System in particular, as well as to predict the likelihood of finding Jupiter, Neptune, and Earth analogues around other stars. Aims. We aim to precisely determine the radii and dynamical masses of transiting planets orbiting the young M star AU Mic using public photometric and spectroscopic datasets. Methods. We performed a joint fit analysis of the TESS and CHEOPS light curves and more than 400 high-resolution spectra collected with several telescopes and instruments. We characterise the stellar activity and physical properties (radius, mass, density) of the transiting planets in the young AU Mic system through joint transit and radial velocity fits with Gaussian processes. Results. We determine a radius of Rpb = 4.79 ± 0.29 R⊕, a mass of Mpb = 9.0 ± 2.7 M⊕, and a bulk density of ρpb = 0.49 ± 0.16 g cm‑3 for the innermost transiting planet AU Mic b. For the second known transiting planet, AU Mic c, we infer a radius of Rpc = 2.79 ± 0.18 R⊕, a mass of Mpc = 14.5 ± 3.4 M⊕, and a bulk density of ρpc = 3.90 ± 1.17 g cm‑3. According to theoretical models, AU Mic b may harbour an H2 envelope larger than 5% by mass, with a fraction of rock and a fraction of water. AU Mic c could be made of rock and/or water and may have an H2 atmosphere comprising at most 5% of its mass. AU Mic b has retained most of its atmosphere but might lose it over tens of millions of years due to the strong stellar radiation, while AU Mic c likely suffers much less photo-evaporation because it lies at a larger separation from its host. Using all the datasets in hand, we determine a 3σ upper mass limit of Mp[d] sin i = 8.6 M⊕ for the AU Mic'd' TTV-candidate. In addition, we do not confirm the recently proposed existence of the planet candidate AU Mic 'e' with an orbital period of 33.4 days. We investigated the level of the radial velocity variations and show that it is lower at longer wavelength with smaller changes from one observational campaign to another.