  We present a spectroscopic analysis of the GIRAFFE and UVES data collected by
the Gaia-ESO survey for the young open cluster NGC 3293. Archive spectra from
the same instruments obtained in the framework of the `VLT-FLAMES survey of
massive stars' are also analysed. Atmospheric parameters, non-LTE chemical
abundances for six elements, or variability information are reported for a
total of about 160 B stars spanning a wide range in terms of spectral types (B1
to B9.5) and rotation rate (up to 350 km/s). We take advantage of the
multi-epoch observations to detect several binary systems or intrinsically
line-profile variables. A deconvolution algorithm is used to infer the current,
true (deprojected) rotational velocity distribution. We find a broad,
Gaussian-like distribution peaking around 200-250 km/s. Although some stars
populate the high-velocity tail, most stars in the cluster appear to rotate far
from critical. We discuss the chemical properties of the cluster, including the
low occurrence of abundance peculiarities in the late B stars and the paucity
of objects showing CN-cycle burning products at their surface. We argue that
the former result can largely be explained by the inhibition of diffusion
effects because of fast rotation, while the latter is generally in accord with
the predictions of single-star evolutionary models under the assumption of a
wide range of initial spin rates at the onset of main-sequence evolution.
However, we find some evidence for a less efficient mixing in two quite rapidly
rotating stars that are among the most massive objects in our sample. Finally,
we obtain a cluster age of ~20 Myrs through a detailed, star-to-star correction
of our results for the effect of stellar rotation. This is significantly older
than previous estimates from turn-off fitting that fully relied on classical,
non-rotating isochrones. [abridged]
