  Protons and antiprotons at collider energies are a source of high energy
Weizs\"acker--Williams photons. This may open a possibility to study exclusive
photoproduction of heavy vector mesons at energies much larger than possible at
the HERA accelerator. Here we present a detailed investigation of the exclusive
$J/\psi$ photoproduction in proton-proton (RHIC, LHC) and proton-antiproton
(Tevatron) collisions. We calculate several differential distributions in $t_1,
t_2, y, \phi$, as well as transverse momentum distributions of $J/\Psi$'s. We
discuss correlations in the azimuthal angle between outgoing protons or proton
and antiproton as well as in the ($t_1, t_2$) space. Differently from
electroproduction experiments, here both colliding beam particles can be a
source of photons, and we find large interference terms in azimuthal angle
distributions in a broad range of rapidities of the produced meson. We also
include the spin--flip parts in the electromagnetic vertices. We discuss the
effect of absorptive corrections on various distributions. Interestingly,
absorption corrections induce a charge asymmetry in rapidity distributions, and
are larger for $p p$ reactions than for the $p \bar p$ case. The reaction
considered here constitutes an important nonreduceable background in recently
proposed searches for odderon exchange.
