We forecast constraints on cosmological parameters with primary cosmic microwave background (CMB) anisotropy information and weak lensing reconstruction with a future post-Planck CMB experiment, the Cosmic Origins Explorer (COrE), using oscillation data on the neutrino mass splittings as prior information. Our Markov chain Monte Carlo (MCMC) simulations in flat models with a non-evolving equation of state of dark energy w give typical 68 per cent upper bounds on the total neutrino mass of 0.136 and 0.098 eV for the inverted and normal hierarchies, respectively, assuming the total summed mass is close to the minimum allowed by the oscillation data for the respective hierarchies (0.10 and 0.06 eV). Including geometric information from future baryon acoustic oscillation measurements with the complete Baryon Oscillation Spectroscopic Survey, Type Ia supernovae distance moduli from Wide-Field Infrared Survey Telescope (WFIRST) and a realistic prior on the Hubble constant, these upper limits shrink to 0.118 and 0.080 eV for the inverted and normal hierarchies, respectively. Addition of these distance priors also yields per cent-level constraints on w. We find tension between our MCMC results and the results of a Fisher matrix analysis, most likely due to a strong geometric degeneracy between the total neutrino mass, the Hubble constant and w in the unlensed CMB power spectra. If the minimal-mass, normal hierarchy were realized in nature, the inverted hierarchy should be disfavoured by the full data combination at typically greater than the 2σ level. For the minimal-mass inverted hierarchy, we compute the Bayes factor between the two hierarchies for various combinations of our forecast data sets, and find that the future cosmological probes considered here should be able to provide ‘strong’ evidence (odds ratio 12:1) for the inverted hierarchy. Finally, we consider potential biases of the other cosmological parameters from assuming the wrong hierarchy and find that all biases on the parameters are below their 1σ marginalized errors.