  We compute the decay rates for the processes $Z\to V+\gamma$, where $Z$ is
the $Z$ boson, $\gamma$ is the photon, and $V$ is one of the vector quarkonia
$J/\psi$ or $\Upsilon(nS)$, with $n=1$, $2$, or $3$. Our computations include
corrections through relative orders $\alpha_s$ and $v^2$ and resummations of
logarithms of $m_Z^2/m_Q^2$, to all orders in $\alpha_s$, at NLL accuracy. ($v$
is the velocity of the heavy quark $Q$ or the heavy antiquark $\bar{Q}$ in the
quarkonium rest frame, and $m_Z$ and $m_Q$ are the masses of $Z$ and $Q$,
respectively.) Our calculations are the first to include both the
order-$\alpha_s$ correction to the light-cone distributions amplitude and the
resummation of logarithms of $m_Z^2/m_Q^2$ and are the first calculations for
the $\Upsilon(2S)$ and $\Upsilon(3S)$ final states. The resummations of
logarithms of $m_Z^2/m_Q^2$ that are associated with the order-$\alpha_s$ and
order-$v^2$ corrections are carried out by making use of the Abel-Pad\'e
method. We confirm the analytic result for the order-$v^2$ correction that was
presented in a previous publication, and we correct the relative sign of the
direct and indirect amplitudes and some choices of scales in that publication.
Our branching fractions for $Z\to J/\psi+\gamma$ and $Z\to \Upsilon(1S)+\gamma$
differ by $2.0\,\sigma$ and $-4.0\,\sigma$, respectively, from the branching
fractions that are given in the most recent publication on this topic (in units
of the uncertainties that are given in that publication). However, we argue
that the uncertainties in the rates are underestimated in that publication.
