Aims: We perform a comprehensive multiwavelength analysis of a sample of 20 starburst galaxies that show a substantial population of Wolf-Rayet (WR) stars. In this paper, the third of the series, we present the analysis of the O and WR star populations within these galaxies. Methods: We study the spatial localization of the WR-rich clusters via the detection of the blue WR bump, mainly composed by the broad He ii λ4686 line and attributed to WN stars, and the red WR bump, composed by the broad C iv λ5808 line observed in WC stars. We perform a detailed fitting of the nebular and broad emission lines within these broad features and derive the numbers of WN, WC and O stars using (i) the standard assumption of constant WR luminosities and (ii) considering metallicity-dependent WR luminosities. We then compare our results with the predictions given by evolutionary synthesis models and with previous empirical results. Results: We report the detection of blue WR bump in 20 regions, but the red WR bump is only detected in six. Aperture effects and the exact positioning of the slit onto the WR-rich bursts play a fundamental role in their detection. The nebular He ii λ4686 line is detected in 21 regions; its intensity clearly decreases with increasing metallicity. We derive an empirical estimation of the WNL/(WNL+O) ratio using the intensity of the broad He ii line assuming metallicity-dependent WR luminosities. As expected, the total number of WR stars increases with increasing metallicity, but objects with 12 + log(O/H) < 8.2 show a rather constant WR/(WR+O) ratio. The computed WCE/WNL ratios are different than those empirically found in nearby star-forming galaxies, indicating that the observed galaxies are experiencing a strong and very short burst. Considering metallicity-dependent WR luminosities, our data agree with a Salpeter-like IMF in all regimes. Conclusions: We consider that the contribution of the WCE stars is not negligible at low metallicities, but deeper observations are needed to detect the red WR bump because of the decreasing of the WR luminosities with decreasing metallicity. Although available models reproduce the WR properties at high metallicities fairly well, new evolutionary synthesis models for young starbursts including all involved parameters (age, metallicity, star-formation history, IMF and WR stars properties such as metallicity-dependent WR luminosities, stellar rotation and the WR binnary channel) are absolutely needed to perform an appropriate comparison with the observational data. Based on observations made with NOT (Nordic Optical Telescope), INT (Isaac Newton Telescope) and WHT (William Herschel Telescope) operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway and Sweden (NOT) or the Isaac Newton Group (INT, WHT) in the Spanish Observatorio del Roque de Los Muchachos of the Instituto de Astrofísica de Canarias.