In response to increasing human emissions, the global ocean is continually warming. The spatial distribution of this warming can result from several mechanisms, difficult to disentangle in observations. Idealized modelling studies have successfully separated the contribution of additional heat passively entering the ocean, from the contribution of the changing circulation redistributing the pre-existing heat in response to perturbations in air-sea fluxes. However, the timescales of these different contributions have been largely unexplored so far. Here, we revisit this decomposition with a novel numerical framework to investigate the mechanisms driving regional ocean warming and its emergence from internal variability. Based on the IPSL-CM6A-LR coupled model and its large ensemble of transient climate change simulations, we extract both the internal fluctuations and the externally-forced signal in each component of the surface fluxes. With a stand-alone configuration of the ocean, we then test the response to perturbations applied on all surface fluxes together or individually. We find that the contribution of the different processes can largely vary in time, reinforcing or counteracting each other, causing the time of emergence of subsurface temperature changes to be advanced or delayed. Anthropogenic warming in the upper ocean water-masses is generally driven by the uptake of excess heat passively stored by the ocean circulation. Circulation changes have a minor role at the time when these signals emerge. On the contrary, in the deeper ocean, circulation changes are much more sensitive to surface forcings and play an important role in setting the timescales of ocean warming, through redistributive warming or cooling.