Context. The double cluster NGC 1850 in the Large Magellanic Cloud is the nearest young massive cluster of the Local Group with a mass similar to those of Galactic globular clusters. Recent studies have revealed an extended morphology of its main-sequence turn-off (MSTO), which can be interpreted as a spread in either age or internal rotation. In spite of its proximity, an accurate spectroscopic determination of its chemical properties is still missing. Aims. We aim to investigate the general chemistry and the kinematics of this stellar system to test whether possible signs of multiple populations are observable in this cluster. Methods. We analysed the spectra obtained with MUSE in adaptive optics mode of 1167 stars in both components of this cluster (NGC 1850A and NGC 1850B). Thanks to this data set, we were able to measure accurate global metallicities, Ba abundances, and radial velocities for a sample of 38 red supergiants and a guess of the oxygen abundance in the brightest turn-off stars belonging to NGC 1850A. Results. We find an average metallicity of ⟨[M/H]⟩ = −0.31 ± 0.01, a mean Ba abundance of ⟨[Ba/Fe]⟩ = +0.40 ± 0.02, and a systemic radial velocity of ⟨vLOS⟩ = 251.1 ± 0.3 km s−1. The dispersion of the radial velocities suggests a dynamical mass of log(M/M⊙) = 4.84 ± 0.1, while no significant systemic rotation is detected. We detect a significant bimodality in O I line strength among the turn-off stars of NGC 1850A with ∼66% of stars with [O/Fe]∼ − 0.16 and the rest with no detectable line. The majority of O-weak stars preferentially populate the red side of the MSTO and show H lines in emission, suggesting that they are Be stars rotating close to their critical velocity. Among normal MSTO stars, red stars have broader line profiles than blue ones, on average, suggesting a correlation between colour and rotational velocity. Conclusions. The mean metallicity of this cluster lies at the metal-rich side of the metallicity distribution of the Large Magellanic Cloud following its age-metallicity relation. The Ba and O abundances agree with those measured in the bar of this galaxy. The correlation between line broadening and colour suggests that the observed colour spread among turn-off stars can be due to a wide range in rotational velocity covered by these stars.