  In this letter we complement previous studies on exclusive vector meson
photoproduction in hadronic collisions presenting a comprehensive analysis of
the $t$ - spectrum measured in exclusive $\rho$ and $J/\Psi$ photoproduction in
$pA$ collisions at the LHC. We compute the differential cross sections
considering two phenomenological models for the gluon saturation effects and
present predictions for $pPb$ and $pCa$ collisions. Moreover, we compare our
predictions with the recent preliminary CMS data for the exclusive $\rho$
photoproduction. We demonstrate that the gluon saturation models are able to
describe the CMS data at small - $t$. On the other hand, the models
underestimate the few data point at large -- $t$. Our results indicate that
future measurements of the large -- $t$ region can be useful to probe the
presence or absence of a dip in the $t$ -- spectrum and discriminate between
the different approaches to the gluon saturation effects.
