In this paper we recast the analysis of twofold saddle point variational formulations for several nonlinear boundary value problems arising in continuum mechanics, and derive reliable and efficient residual-based a posteriori error estimators for the associated Galerkin schemes. We illustrate the main results with nonlinear elliptic equations modelling heat conduction and hyperelasticity. The main tools of our analysis include a global inf-sup condition for a linearization of the problem, Helmholtz's decompositions, local approximation properties of the Raviart-Thomas and Clement interpolation operators, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions. Finally, several numerical results confirming the theoretical properties of the estimator and showing the behaviour of the associated adaptive algorithms are provided.