  Using the framework that interpolates between the leading power limit of the
Color Glass Condensate and the High Energy (or $k_{T}$) factorization we
calculate the direct component of the forward dijet production in
ultra-peripheral $\mathrm{Pb}$-$\mathrm{Pb}$ collisions at CM energy
$5.1\,\mathrm{TeV}$ per nucleon pair. The formalism is applicable when the
average transverse momentum of the dijet system $P_{T}$ is much bigger than the
saturation scale $Q_{s}$, $P_{T}\gg Q_{s}$, while the imbalance of the dijet
system can be arbitrary. The cross section is uniquely sensitive to the
Weizs\"acker-Williams (WW) unintegrated gluon distribution, which is far less
known from experimental data than the most common dipole gluon distribution
appearing in inclusive small-$x$ processes. We have calculated cross sections
and nuclear modification ratios using WW gluon distribution obtained from the
dipole gluon density through the Gaussian approximation. The dipole gluon
distribution used to get WW was fitted to the inclusive HERA data with the
nonlinear extension of unified BFKL+DGLAP evolution equation. The saturation
effects are visible but rather weak for realistic $p_{T}$ cut on the dijet
system, reaching about $20\%$ with the cut as low as $6\,\mathrm{GeV}$. We find
that the LO collinear factorization with nuclear leading twist shadowing
predicts quite similar effects.
