We analyze the component structure of models for 4D N= 1 supersymmetric nonlinear electrodynamics that enjoy invariance under continuous duality rotations. The N= 1 supersymmetric Born-Infeld action is a member of this family. Such dynamical systems have a more complicated structure, especially in the presence of supergravity, as compared with well-studied effective supersymmetric theories containing at most two derivatives (including nonlinear Kähler sigma-models). As a result, when deriving their canonically normalized component actions, it becomes impractical and cumbersome to follow the traditional approach of (i) reducing to components; and then (ii) applying a field-dependent Weyl and local chiral transformation. It proves to be more efficient to follow the Kugo-Uehara scheme which consists of (i) extending the superfield theory to a super-Weyl invariant system; and then (ii) applying a plain component reduction along with imposing a suitable super-Weyl gauge condition. Here we implement this scheme to derive the bosonic action of self-dual supersymmetric electrodynamics coupled to the dilaton-axion chiral multiplet and a Kähler sigma-model. In the fermionic sector, the action contains higher derivative terms. In the globally supersymmetric case, a nonlinear field redefinition is explicitly constructed which eliminates all the higher derivative terms and brings the fermionic action to a one-parameter deformation of the Akulov-Volkov action for the Goldstino. The Akulov-Volkov action emerges, in particular, in the case of the N= 1 supersymmetric Born-Infeld action.