Very recent experimental information obtained from the Belle experiment, along with that accumulated by the BABAR and LHCb experiments, has shown the existence of anomalies in the ratios $R(D)$ and $R(D^{*})$ associated with the charged-current transition $b \to c \tau \bar{\nu}_\tau$. Although the Belle measurements are in agreement with the standard model (SM) predictions, the new experimental world averages still exhibit a tension. In addition, the $D^\ast$ longitudinal polarization $F_L(D^\ast)$ related with the channel $B \to D^\ast \tau \bar{\nu}_\tau$ observed by the Belle Collaboration and the ratio $R(J/\psi)$ measured by the LHCb Collaboration also show discrepancies with their corresponding SM estimations. We present a model-independent study based on the most general effective Lagrangian that yields a tree-level effective contribution to the transition $b \to c \tau \bar{\nu}_\tau$ induced by a general $W^\prime$ boson. Instead of considering any specific new physics (NP) realization, we perform an analysis by considering all of the different chiral charges to the charm-bottom and $\tau$-$\nu_{\tau}$ interaction terms with a charged $W^\prime$ boson that explain the anomalies. We present a phenomenological study of parameter space allowed by the new experimental $b \to c \tau \bar{\nu}_\tau$ data and with the mono-tau signature $pp \to \tau_h X + \rm{MET}$ at the LHC. For comparison, we include some of the $W^\prime$ boson NP realizations that have already been studied in the literature.