Context. For more than two decades we have known that galaxy morphological segregation is present in the Local Universe. It is important to see how this relation evolves with cosmic time. Aims. To investigate how galaxy assembly took place with cosmic time, we explore the evolution of the morphology-density relation up to redshift z ∼ 1 using about 10 000 galaxies drawn from the zCOSMOS Galaxy Redshift Survey. Taking advantage of accurate HST/ACS morphologies from the COSMOS survey, of the well-characterised zCOSMOS 3D environment, and of a large sample of galaxies with spectroscopic redshift, we want to study here the evolution of the morphology-density relation up to z ∼ 1 and its dependence on galaxy luminosity and stellar mass. The multi-wavelength coverage of the field also allows a first study of the galaxy morphological segregation dependence on colour. We further attempt to disentangle between processes that occurred early in the history of the Universe or late in the life of galaxies. Methods. The zCOSMOS field benefits of high-resolution imaging in the F814W filter from the Advanced Camera for Survey (ACS). We use standard morphology classifiers, optimised for being robust against band-shifting and surface brightness dimming, and a new, objective, and automated method to convert morphological parameters into early, spiral, and irregular types. We use about 10 000 galaxies down to I AB = 22.5 with a spectroscopic sampling rate of 33% to characterise the environment of galaxies up to z ∼ 1 from the 100 kpc scales of galaxy groups up to the 100 Mpc scales of the cosmic web. The evolution of the morphology-density relation in different environments is then studied for luminosity and stellar-mass selected, volume-limited samples of galaxies. The trends are described and related to the various physical processes that could play a relevant role in the build-up of the morphology-density relation. Results. We confirm that the morphological segregation is present up to z ∼ 1 for luminosity-selected, volume-limited samples. The behaviour of the morphology-density relation gets flatter at fixed masses expecially above 10 10.6 M . We suggest the existence of a critical mass above which the physical processes governing galaxy stellar mass also determine the shaping of the galaxy more than its environment. We finally show that at a fixed morphology there is still a residual variation in galaxy colours with density. Conclusions. The observed evolution with redshift of the morphology-density relation offers an opportunity to trace the effect of nature and nurture as a function of environment. Even though it is based mainly on a biased view, the environmental dependence of the morphological evolution for luminosity-selected, volume-limited samples seems to indicate that nurture is in play. On the other hand, the lack of evolution observed for earlytype and spiral galaxies that are more massive than 10 10.8 M independents of the environment indicates that nature has imprinted these properties early in the life of...