Protons and antiprotons at collider energies are a source of high-energy Weizsaecker-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 (Relativistic Heavy Ion Collider, CERN LHC) and proton-antiproton (Tevatron) collisions. We calculate several differential distributions in t{sub 1}, t{sub 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{sub 1},t{sub 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 pp reactions than for the pp case. The reaction considered here constitutes an important nonreduceable background in recently proposed searches for odderon exchange.