When a deposited layer of granular material fully immersed in a liquid is suddenly inclined above a certain critical angle, it starts to flow down the slope. The initial dynamics of these underwater avalanches strongly depends on the initial volume fraction. If the granular bed is initially loose, i.e. looser than the critical state, the avalanche is triggered almost instantaneously and exhibits a strong acceleration, whereas for an initially dense granular bed, i.e. denser than the critical state, the avalanche's mobility remains low for some time before it starts flowing normally. This behaviour can be explained by a combination of geometrical granular dilatancy and pore pressure feedback on the granular media. In this contribution, a continuum formulation is presented and implemented in a three-dimensional continuum numerical model. The originality of the present model is to incorporate dilatancy as an elasto-plastic normal stress or pressure and not as a modification of the friction coefficient. This allows an explanation of the two different behaviours of initially loose and dense underwater avalanches. It also highlights the contribution from each depth-resolved variable in the strongly coupled transition to a flowing avalanche. The model compares favourably with existing experiments for the initiation of underwater granular avalanches. Results reveal the interplay between shear-induced changes of the granular stress and fluid pressure in the dynamics of avalanches. The characteristic time of the triggering phase is nearly independent of the local rheological parameters, whereas the initial drop in pore pressure and the surface velocity at steady state still strongly depend on them. Finally, the multidimensional capabilities of the model are illustrated for the two-dimensional Hele-Shaw configuration and some of the observed differences between one-dimensional simulations and experiments are clarified.