  We discuss double-diffractive (double-elastic) production of the
$\eta'$-meson in the $pp \to p\eta'p$ reaction within the formalism of
unintegrated gluon distribution functions (UGDF). We estimate also the
contribution of $\gamma^* \gamma^* \to \eta'$ fusion. The distributions in the
Feynman $x_F$ (or rapidity), transferred four-momenta squared between initial
and final protons ($t_1$, $t_2$) and azimuthal angle difference between
outgoing protons ($\Phi$) are calculated. The deviations from the
$\sin^2(\Phi)$ dependence predicted by one-step vector-vector-pseudoscalar
coupling are quantified and discussed. The results are compared with the
results of the WA102 collaboration at CERN. Most of the models of UGDF from the
literature give too small cross section as compared to the WA102 data and
predict angular distribution in relative azimuthal angle strongly asymmetric
with respect to $\pi/2$ in disagreement with the WA102 data. This points to a
different mechanism at the WA102 energy. Predictions for RHIC, Tevatron and LHC
are given. We find that the normalization, $t_{1,2}$ dependences as well as
deviations from $\sin^2(\Phi)$ of double-diffractive double-elastic cross
section are extremely sensitive to the choice of UGDF. Possible implications
for UGDFs are discussed.
