Context. Accurate stellar parameters of individual objects in binary systems are essential to constrain the effects of binarity on stellar evolution. These parameters serve as a prerequisite to probing existing and future theoretical evolutionary models. Aims. We aim to derive the atmospheric parameters of the 31 double-lined spectroscopic binaries in the Tarantula Massive Binary Monitoring sample. This sample, composed of detached, semi-detached and contact systems with at least one of the components classified as an O-type star, is an excellent test-bed to study how binarity can impact our knowledge of the evolution of massive stars. Methods. In the present paper, 32 epochs of FLAMES/GIRAFFE spectra are analysed by using spectral disentangling to construct the individual spectra of 62 components. We then apply the CMFGEN atmosphere code to determine their stellar parameters and their helium, carbon, and nitrogen surface abundances. Results. Among the 31 systems that we study in the present paper, we identify between 48 and 77% of them as detached, likely pre-interacting systems, 16% as semi-detached systems, and between 5 and 35% as systems in or close to contact phase. Based on the properties of their components, we show that the effects of tides on chemical mixing are limited. Components on longer-period orbits show higher nitrogen enrichment at their surface than those on shorter-period orbits, in contrast to expectations of rotational or tidal mixing, implying that other mechanisms play a role in this process. For semi-detached systems, components that fill their Roche lobe are mass donors. They exhibit higher nitrogen content at their surface and rotate more slowly than their companions. By accreting new material, their companions spin faster and are likely rejuvenated. Their locations in the N − v sin i diagram tend to show that binary products are good candidates to populate the two groups of stars (slowly rotating, nitrogen-enriched objects and rapidly rotating non-enriched objects) that cannot be reproduced through single-star population synthesis. Finally, we find no peculiar surface abundances for the components in (over-)contact systems, as has been suggested by evolutionary models for tidal mixing. Conclusions. This sample, consisting of 31 massive binary systems, is the largest sample of binaries composed of at least one O-type star to be studied in such a homogeneous way by applying spectral disentangling and atmosphere modelling. The study of these objects gives us strong observational constraints to test theoretical binary evolutionary tracks.