We investigate the structure of the one- and two-nucleon axial-current operators necessary for the partial conservation of the nuclear axial-current elastic matrix element in the one-boson-exchange approximation to the two-nucleon Bethe-Salpeter equation. We use three models for this purpose: (a) the linear sigma model, (b) the nonlinear sigma model, (c) a hybrid model, which is, roughly speaking, a linear combination of (a) and (b). We construct a partially conserved nuclear axial-current elastic matrix element in models (a) and (c) provided that the associated nuclear wave functions are solutions to the Bethe-Salpeter equation with a potential made of one-boson-exchange diagrams. In the nonlinear sigma model the nuclear one-body axial current is partially conserved by itself, without reference to the nuclear wave function, whereas the two-body axial current partial conservation is violated by terms of order 1/${\mathit{f}}_{\mathrm{\ensuremath{\pi}}}^{2}$. The complete axial current in models (a) and (c) and the one-body axial current in model (b) are applicable to the construction of the deuteron electroweak process amplitudes, for example. The divergence of the axial-current matrix element is proportional to the pion absorption nuclear matrix element, which leads to another potential application in the foundation of chiral perturbation theory for pion-two-nucleon processes. Consistency between the nuclear axial-currents and the underlying nuclear dynamics in models (a) and (c) is a new condition imposed by the partial conservation of the nuclear axial-current matrix element. We also examine conditions imposed on the form of the nucleon self-energy by the nucleon and meson axial Ward-Takahashi identities, as well as the approximations that satisfy the said conditions. We show that besides the first Born approximation, the so-called Hartree+random-phase approximation satisfies chiral Ward-Takahashi idenitities in models (a) and (c). \textcopyright{} 1996 The American Physical Society.