It is shown that an account for the final-state interaction of real or virtual nucleon and antinucleon produced in the processes $J/\psi \to p\overline{p}\gamma$, $\psi \left(2S\right)\to p\overline{p}\gamma$, $J/\psi \to p\overline{p}\omega$, and $J/\psi \to 3\left({\pi }^{+}{\pi }^{-}\right)\gamma$ near the threshold of $N\overline{N}$ pair production allows one to obtain self-consistent description of these processes. Predictions of our model are in good agreement with experimental data available. The proposed potential model also reproduces the corresponding partial cross sections of $p\overline{p}$ scattering.