We studied polarisation-dependent exclusive $J/\psi$ production in ultraperipheral heavy-ion collisions at RHIC and LHC energies and in $eA$ collisions at EIC energies, in the framework of color glass condensate effective theory. The azimuthal averaged $J/\psi$ production cross section measured by STAR and ALICE is accurately described by our calculation. We further predict significant cos 2$\phi$ azimuthal asymmetries both in UPCs and in $eA$ collisions, which are attributed to the linearly polarized photon, the double-slit interference effect in photonuclear reactions, and the final state soft photon radiation effect. This study may provide further information on limiting gluon transverse spatial distribution inside large nuclei.