A census of the Wolf-Rayet content in Westerlund 1 from near-infrared imaging and spectroscopy

New Technology Telescope (NTT)/Son of Isaac (SOFI) imaging and spectroscopy of the Wolf-Rayet population in the massive cluster Westerlund 1 are presented. Narrow-band near-infrared (IR) imaging together with follow up spectroscopy reveals four new Wolf-Rayet stars, of which three were independently identified recently by Groh et al., bringing the confirmed Wolf-Rayet content to 24 (23 excluding source S) - representing 8 per cent of the known Galactic Wolf-Rayet population - comprising eight WC stars and 16 (15) WN stars. Revised coordinates and near-IR photometry are presented, whilst a quantitative near-IR spectral classification scheme for Wolf-Rayet stars is presented and applied to members of Westerlund 1. Late subtypes are dominant, with no subtypes earlier than WN5 or WC8 for the nitrogen and carbon sequences, respectively. A qualitative inspection of the WN stars suggests that most (similar to 75 per cent) are highly H deficient. The Wolf-Rayet binary fraction is high (>= 62 per cent), on the basis of dust emission from WC stars, in addition to a significant WN binary fraction from hard X-ray detections according to Clark et al. We exploit the large WN population of Westerlund 1 to reassess its distance (similar to 5.0 kpc) and extinction (A(KS) similar to 0.96 mag), such that it is located at the edge of the Galactic bar, with an oxygen metallicity similar to 60 per cent higher than Orion. The observed ratio of WR stars to red and yellow hypergiants, N(WR)/N(RSG + YHG) similar to 3, favours an age of similar to 4.5-5.0 Myr, with individual Wolf-Rayet stars descended from progenitors of initial mass similar to 40-55 M-circle dot. Qualitative estimates of current masses for non-dusty, H-free WR stars are presented, revealing 10-18 M-circle dot, such that similar to 75 per cent of the initial stellar mass has been removed via stellar winds or close binary evolution. We present a revision to the cluster turn-off mass for other Milky Way clusters in which Wolf-Rayet stars are known, based upon the latest temperature calibration for OB stars. Finally, comparisons between the observed WR population and subtype distribution in Westerlund 1 and instantaneous burst evolutionary synthesis models are presented.