  The prospects are presented for precise measurements of the branching ratios
of the purely leptonic $B_c^+ \to \tau^+ \nu_\tau$ and $B^+ \to \tau^+
\nu_\tau$ decays at the Future Circular Collider (FCC). This work is focused on
the hadronic $\tau^{+} \to \pi^+ \pi^+ \pi^- \bar{\nu}_\tau$ decay in both
$B_c^+ \to \tau^+ \nu_\tau$ and $B^+ \to \tau^+ \nu_\tau$ processes. Events are
selected with two Boosted Decision Tree algorithms to optimise the separation
between the two signal processes as well as the generic hadronic $Z$ decay
backgrounds. The range of the expected precision for both signals are evaluated
in different scenarios of non-ideal background modelling. This paper
demonstrates, for the first time, that the $B^+ \to \tau^+ \nu_\tau$ decay can
be well separated from both $B_c^+ \to \tau^+ \nu_\tau$ and generic $Z\to
b\bar{b}$ processes in the FCC-ee collision environment and proposes the
corresponding branching ratio measurement as a novel way to determine the CKM
matrix element $|V_{ub}|$. The theoretical impacts of both $B^+ \to \tau^+
\nu_\tau$ and $B_c^+ \to \tau^+ \nu_\tau$ measurements on New Physics cases are
discussed for interpretations in the generic Two-Higgs-doublet model and
leptoquark models.
