We investigate the influence of the direct and converse piezoelectric effect, i.e., electromechanical coupling (EC), on the strain and the built-in electric field in hexagonal group-III nitride heterostructures under hydrostatic pressure. Particularly, we derive the analytic formulas for pressure dependences of the strain tensor components and the built-in electric field in wurtzite heterostructures, taking into account the EC effect. Then, we calculate pressure coefficients of the light emission, $d{E}_{E}∕dP$, in various $\mathrm{Ga}\mathrm{N}∕\mathrm{Al}\mathrm{Ga}\mathrm{N}$ and $\mathrm{In}\mathrm{Ga}\mathrm{N}∕\mathrm{Ga}\mathrm{N}$ superlattices and quantum wells (QWs). Generally, our calculations reveal that taking into account the EC leads to the decrease of the pressure derivative of the built-in electric field in the QW region, which further causes an increase of the $d{E}_{E}∕dP$. The contribution of the EC to $d{E}_{E}∕dP$ depends significantly on the geometry, composition, and strain state of heterostructures. We have found that the largest influence of the EC on $d{E}_{E}∕dP$ is for $\mathrm{Ga}\mathrm{N}∕\mathrm{Al}\mathrm{N}$ heterostructures. In $\mathrm{Ga}\mathrm{N}∕\mathrm{Al}\mathrm{Ga}\mathrm{N}$ structures, the contribution of the EC to $d{E}_{E}∕dP$ grows with an increasing well width and Al concentration in the barriers. A larger influence of the EC on $d{E}_{E}∕dP$ is observed for the structures with strained barriers than with strain wells. Interestingly, for $\mathrm{In}\mathrm{Ga}\mathrm{N}∕\mathrm{Ga}\mathrm{N}$ heterostructures grown coherently on GaN substrates, the effect of the EC on $d{E}_{E}∕dP$ is negligible.