Aims. Recent observations have challenged our understanding of rotational mixing in massive stars by revealing a population of fastrotating objects with apparently normal surface nitrogen abundances. However, several questions have arisen because of a number of issues, which have rendered a reinvestigation necessary; these issues include the presence of numerous upper limits for the nitrogen abundance, unknown multiplicity status, and a mix of stars with different physical properties, such as their mass and evolutionary state, which are known to control the amount of rotational mixing. Methods. We have carefully selected a large sample of bright, fast-rotating early-type stars of our Galaxy (40 objects with spectral types between B0.5 and O4). Their high-quality, high-resolution optical spectra were then analysed with the stellar atmosphere modelling codes DETAIL/SURFACE or CMFGEN, depending on the temperature of the target. Several internal and external checks were performed to validate our methods; notably, we compared our results with literature data for some well-known objects, studied the effect of gravity darkening, or confronted the results provided by the two codes for stars amenable to both analyses. Furthermore, we studied the radial velocities of the stars to assess their binarity.Results. This first part of our study presents our methods and provides the derived stellar parameters, He, CNO abundances, and the multiplicity status of every star of the sample. It is the first time that He and CNO abundances of such a large number of Galactic massive fast rotators are determined in a homogeneous way.