  We prove the existence of a universal braided compact quantum group acting on
a graph $\mathrm{C}^*$-algebra in the category of
$\mathbb{T}$-$\mathrm{C}^*$-algebras with a twisted monoidal structure, in the
spirit of the seminal work of S. Wang. To achieve this, we construct a braided
analogue of the free unitary quantum group and study its bosonization. As a
concrete example, we compute this universal braided compact quantum group for
the Cuntz algebra.
