We consider the acceleration of leptons up to GeV-TeV energies in the bow shock wind nebula of PSR J0437-4715 and their subsequent diffusion through the interstellar magnetic fields. The leptons accelerated at the pulsar wind termination surface are injected into re-acceleration in colliding shock flows. Modeled spectra of synchrotron emission from the accelerated electrons and positrons are consistent with the far-ultraviolet and X-ray observations of the nebula carried out with the Hubble Space Telescope and Chandra X-ray Observatory. These observations are employed to constrain the absolute fluxes of relativistic leptons, which are escaping from the nebula and eventually reaching the solar system after energy-dependent diffusion through the local interstellar medium accompanied by synchrotron and Compton losses. It is shown that accelerated leptons from the nebula of PSR J0437-4715 can be responsible both for the enhancement of the positron fraction above a few GeV detected by PAMELA and AMS-02 spectrometers and for the TeV range lepton fluxes observed with H.E.S.S., VERITAS, Fermi, CALET, and DAMPE.