  After leaving the main sequence, massive stars undergo complex evolution,
still poorly understood. With a population of 100s OB stars, the starburst
cluster Westerlund~1 offers an unparallelled environment to study their
evolutionary tracks. We used the VLT/FORS2 to obtain intermediate-resolution
spectroscopy over the range 5800-9000A of ~ 100 likely members. We developed
criteria for their spectral classification using only spectral features in the
range observed. We discuss these criteria, useful for spectral classification
of early-type stars in the GAIA spectral region, in the appendix. Using these
criteria, we obtain spectral classifications, probably accurate to one subtype,
for 57 objects, most of which had no previous classification or a generic
classification. We identify more than 50 objects as OB supergiants. We find
almost 30 luminous early-B supergiants and a number of less luminous late-O
supergiants. In addition, we find a few mid B supergiants with very high
luminosity, some of them displaying signs of heavy mass loss. All these stars
form a sequence compatible with theoretical evolutionary tracks. In addition,
two early B supergiants also show indication of heavy mass loss and may
represent the evolutionary phase immediately prior to the Wolf-Rayet stage. We
investigate cluster properties using the spectral types and existing
photometry. We find that the reddening law to the cluster does not deviate
strongly from standard, even though extinction is quite variable, with an
average value A_v=10.8. Though evolutionary tracks for high-mass stars are
subject to large uncertainties, our data support an age of >~5Myr and a
distance ~5kpc for Westerlund 1. The spectral types observed are compatible
with a single burst of star formation (the age range is very unlikely to be
>1Myr).[ABRIDGED]
