  We investigate the influence of an additional bosonic bath onto the real-time
dynamics of a localized orbital coupled to conduction band with an
energy-dependent coupling function $\Gamma(\varepsilon) \propto
|\varepsilon|^r$. Recently, a rich phase diagram has been found in this
Bose-Fermi Anderson model, where the transitions between competing ground
states are governed by quantum critical points. In addition to a transition
between a Kondo singlet and a local moment, a localized phase has been
established once the coupling to a sub-ohmic bosonic bath exceeds a critical
value. Using the time-dependent numerical renormalization group approach, we
show that the non-equilibrium dynamics with F-type of bath exponents can be
fully understand within an effective single-impurity Anderson model using a
renormalized local Coulomb interaction $U_{\rm ren}$. For regimes with B-type
of bath exponents, the nature of the bosonic bath and the coupling strength has
a profound impact on the electron dynamics which can only partially be
understood using an appropriate $U_{\rm ren}$. The local expectation values
always reach a steady state at very long times. By a scaling analysis for
$\Lambda \to 1^+$, we find thermalization of the system only in the strong
coupling regime. In the local moment and in the localized phase significant
deviations between the steady-state value and the thermal equilibrium value are
found that are related to the distance to the quantum critical point.
