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 Weizsacker-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.
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