We study the epicyclic oscillatory motion around circular orbits of the traversable asymptotically flat and reflection-symmetric wormholes obtained in the Einstein-Dirac-Maxwell theory without applying exotic matter in their construction. We determine frequencies of the orbital and epicyclic motion in the Keplerian disks having inner edge at the marginally stable circular geodesic of the spacetime. The obtained frequencies are applied in the so-called geodesic models of high-frequency quasiperiodic oscillations (HF QPOs) observed in microquasars and active galactic nuclei containing a supermassive central object. We show that even the simplest epicyclic resonance variant of the geodesic models can explain the HF QPOs observed in many active galactic nuclei for realistic choices of the wormhole parameters, but there are some of the sources where only wormholes with unrealistically large values of the parameters can be sufficient for the explanation. On the other hand, in the case of microquasars, the observed HF QPOs strongly restrict the acceptable values of the wormhole parameters.