A spectroscopic analysis has been undertaken for the B-type multiple systems (excluding those with supergiant primaries) in the VLT-FLAMES Tarantula Survey (VFTS). Projected rotational velocities, vesini, for the primaries have been estimated using a Fourier Transform technique and confirmed by fitting rotationally broadened profiles. A subset of 33 systems with vesini ≤ 80 km s-1 have been analysed using a TLUSTY grid of model atmospheres to estimate stellar parameters and surface abundances for the primaries. The effects of a potential flux contribution from an unseen secondary have also been considered. For 20 targets it was possible to reliably estimate their effective temperatures (Teff) but for the other 13 objects it was only possible to provide a constraint of 20 000 ≤ Teff ≤ 26 000 K - the other parameters estimated for these targets will be consequently less reliable. The estimated stellar properties are compared with evolutionary models and are generally consistent with their membership of 30 Doradus, while the nature of the secondaries of 3 SB2 system is discussed. A comparison with a sample of single stars with vesini ≤ 80 km s-1 obtained from the VFTS and analysed with the same techniques implies that the atmospheric parameters and nitrogen abundances of the two samples are similar. However, the binary sample may have a lack of primaries with significant nitrogen enhancements, which would be consistent with them having low rotational velocities and having effectively evolved as single stars without significant rotational mixing. This result, which may be actually a consequence of the limitations of the pathfinder investigation presented in this paper, should be considered as a motivation for spectroscopic abundance analysis of large samples of binary stars, with high quality observational data. Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programme 182.D-0222.Tables 6 and 7 are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (http://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/603/A91