We evaluate the transverse momentum spectrum of J/ψ (up to semi-hard momenta) in pA and AA collisions taking into account only the initial state effects, but resumming them to all orders in αs2A1/3. In our previous papers we noticed that cold nuclear matter effects alone could not explain the experimental data on rapidity and centrality dependencies of the J/ψ yield in AA collisions indicating the existence of an additional suppression mechanism. Our present calculations indicate that the discrepancy persists and even increases at semi-hard transverse momenta, implying a significant final state effect on J/ψ production in this kinematical domain. The QCD dipole model we employ is only marginally applicable for J/ψ production at mid-rapidity at RHIC energies but its use is justified in the forward rapidity region. At LHC energies we can quantitatively evaluate the magnitude of cold nuclear matter effects in the entire kinematical region of interest. We present our calculations of J/ψ transverse momentum spectra in pA and AA collisions at LHC and RHIC energies.