  We consider the spin echo dynamics of a nitrogen-vacancy center qubit based
the $S\!= \! 1$ ground state spin manifold, caused by a dynamically polarized
nuclear environment. We show that the echo signal acquires then a nontrivially
time-dependent phase shift. This effect should be observable for polarization
$\approx \! 0.5$ of nuclei within $\sim \! 1$ nm from the qubit, and for the NV
center initialized in a superposition of $m\! = \! 0$ and either $m\! =\! 1$ or
$m\! =\! -1$ states. This phase shift is much smaller when the NV center is
prepared in a superposition of $m\! = \! 1$ and $m\! =\! -1$ states, i.e. when
the qubit couples to the spin environment in a way analogous to that of
spin-$1/2$. For nuclear environment devoid of spins strongly coupled to the
qubit, the phase shift is well described within Gaussian approximation, which
provides an explanation for the dependence of the shift magnitude on the choice
of states on which the qubit is based, and makes it clear that its presence is
related to the linear response of the environment perturbed by an evolving
qubit. Consequently, its observation signifies the presence
environment-mediated self-interaction of the qubit, and hence, it invalidates
the notion that the nuclear environment acts as a source of external noise
driving the qubit. We also show how a careful comparison of the echo signal
from qubits based on $m\! = \! 0,1$ and $m\! =\! \pm 1$ manifolds, can
distinguish between effectively Gaussian and non-Gaussian environment.
