We elaborate an ab initio approach for the evaluation of the one-loop quantum electrodynamical corrections to energy levels of diatomic quasimolecules. The approach accounts for the interaction between an electron and two nuclei in all orders in Zα and can be applied for a wide range of internuclear distances, up to R≈1000 fm. Based on the developed theory, detailed calculations are performed for the self-energy and vacuum-polarization corrections to the energy of the 1σ(g) ground state of the U(92+)-U(91+) dimer that can be produced in slow ion-ion collisions. The calculations predict the remarkable energy shift that arises due to the nonspherical contributions to the electron-nuclei potential taken beyond the standard monopole approximation.