  Quantum quenches to or near criticality give rise to the phenomenon of
\textit{aging}, manifested by glassy-like dynamics at short times and far from
equilibrium. The recent surge of interest in the dynamics of quantum many-body
systems has rejuvenated interest in this phenomenon. Motivated by the
ubiquitous long-range interactions in emerging experimental platforms, it is
vital to study quantum aging in such settings. In this work, we investigate the
dynamical universality and aging in the $d$-dimensional $O(N)$ model with the
long-range coupling $1/x^{d+\sigma}$ and in the mean-field limit $N\to\infty$
that allows an exact treatment. An immediate consequence of long-range coupling
is the emergence of nonlinear light cones. We focus on the correlation and
response functions, and identify a rich scaling behavior depending on how the
corresponding space-time positions are located relative to each other, via a
\textit{local light cone}, and to the time of the quench via a global
\textit{quench light cone}. We determine the initial-slip exponent that governs
the short-time dependence of two-point functions. We highlight the new
qualitative features of aging due to the long-range coupling, in particular in
the region outside the light cones. As an important consequence of long-range
coupling, the correlation function decays as $1/x^{d+\sigma}$ outside the
quench light cone while increasing polynomially with the total time after
quench. This is while, for short time differences, the two-time response
function "equilibrates" at \textit{all} distances even outside this light cone.
Our analytic findings are in excellent agreement with exact numerics, and
provide a useful benchmark for modern experimental platforms with long-range
interactions.
