  We consider the recent LHCb result for $B_c\to J/\psi \tau \nu$ in
conjunction with the existing anomalies in $R(D)$ and $R(D^\star)$ within the
framework of a right-handed current with enhanced couplings to the third
generation. The model predicts a linear relation between the observables and
their SM values in terms of two combinations of parameters. The strong
constraints from $b\to s \gamma$ on $W-W^\prime$ mixing effectively remove one
of the combinations of parameters resulting in an approximate proportionality
between all three observables and their SM values. To accommodate the current
averages for $R(D)$ and $R(D^\star)$, the $W^\prime$ mass should be near 1 TeV,
and possibly accessible to direct searches at the LHC. In this scenario we find
that $R(J/\psi)$ is enhanced by about 20\% with respect to its SM value and
about 1.5$\sigma$ below the central value of the LHCb measurement. The
predicted $d\Gamma/dq^2$ distribution for $B\to D(D^\star) \tau \nu$ is in
agreement with the measurement and the model satisfies the constraint from the
$B_c$ lifetime.
