We extend our general framework for semileptonic decay, originally introduced in N. Penalva et al. [Phys. Rev. D100, 113007 (2019)], with the addition of new physics (NP) tensor terms. In this way, all the NP effective Hamiltonians that are considered in lepton flavour universality violation (LFUV) studies have now been included. Besides, we now also give general expressions that allow for complex Wilson coefficients. The scheme developed is totally general and it can be applied to any charged current semileptonic decay, involving any quark flavors or initial and final hadron states. We show that all the hadronic input, including NP effects, can be parametrized in terms of 16 Lorentz scalar structure functions. In the second part of this work, we use this formalism to obtain the complete NP effects in the \(\Lambda_b\to \Lambda_c \tau\bar\nu_\tau\) semileptonic decay, where LFUV, if finally confirmed, is also expected to be seen. We stress the relevance of the center of mass (CM) \(d^2\Gamma/(d\omega d\cos\theta_\ell)\) and laboratory (LAB) \(d^2\Gamma/(d\omega dE_\ell)\) differential decay widths. While models with very different strengths in the NP terms give the same differential \(d\Gamma/d\omega\) and total decay widths for this decay, they predict very different numerical results for some of the \(\cos\theta_\ell\) and \(E_\ell\) coefficient-functions that determine the above two distributions. Thus, the combined analysis of the CM \(d^2\Gamma/(d\omega d\cos\theta_\ell)\) and LAB \(d^2\Gamma/ (d\omega dE_\ell)\) differential decay widths will help clarifying what kind of NP is a better candidate in order to explain LFUV.