  We develop an efficient variational approach to studying dynamics of a
localized quantum spin coupled to a bath of mobile spinful bosons. We use
parity symmetry to decouple the impurity spin from the environment via a
canonical transformation and reduce the problem to a model of the interacting
bosonic bath. We describe coherent time evolution of the latter using bosonic
Gaussian states as a variational ansatz. We provide full analytical expressions
for equations describing variational time evolution that can be applied to
study in- and out-of-equilibrium phenomena in a wide class of quantum impurity
problems. In the accompanying paper [Y. Ashida {\it et al.}, Phys. Rev. Lett.
123, 183001 (2019)], we present a concrete application of this general
formalism to the analysis of the Rydberg Central Spin Model, in which the
spin-1/2 Rydberg impurity undergoes spin-changing collisions in a dense cloud
of two-component ultracold bosons. To illustrate new features arising from
orbital motion of the bath atoms, we compare our results to the Monte Carlo
study of the model with spatially localized bosons in the bath, in which random
positions of the atoms give rise to random couplings of the standard central
spin model.
