  Taking into account the gluon shadowing and the $J/\psi$ nuclear absorption,
we evaluate the Cold Nuclear Matter (CNM) effects on $J/\psi$ production in
$p$Pb and PbPb collisions at the design LHC energy for PbPb collisions,
$\sqrt{s_{NN}}=5.5$ TeV. In view of the good agreement between the yield
prediction using the LO CSM and the ALICE data at 7 TeV, we employ the latter
model to use a complete description of the kinematics of the underlying $2
\rightarrow 2$ partonic process, namely $g + g \rightarrow J/\psi + g$. We
observe a large $J/\psi$ suppression in $p$Pb and PbPb collisions due to the
strong shadowing of the gluon distribution in the lead ion at the corresponding
gluon momentum fractions. This suppression from CNM effects has a strong
rapidity dependence which may compensate, partially or completely, that of of
the predicted charm recombination in PbPb collisions.
