Context. Charge transfer ( or exchange) reactions between hydrogen atoms and protons in collisionless shocks of supernova remnants (SNRs) are a natural way of producing broad Balmer, Lyman, and other lines of hydrogen. Aims. We wish to quantify the importance of shock-induced, non-thermal hydrogen emission from SNRs in young galaxies. Methods. We present a method estimating the luminosity of broad (similar to 1000 km s(-1)) Ly alpha, Ly beta, Ly gamma., H beta and P alpha lines, as well as the broad and narrow luminosities of the two-photon (2 gamma.) continuum, from existing measurements of the Ha flux. We consider cases of beta = 0.1 and 1, where beta = T(e)/T(p) is the ratio of electron-to-proton temperatures. We examine a modest sample of 8 proximate, Balmer-dominated SNRs from our Galaxy and the Large Magellanic Cloud. The expected broad Lya luminosity per object is at most similar to 10(36) erg s(-1). The 2 gamma. continuum luminosities are comparable to the broad Ha and Lya ones. We restrict our analysis to homogenous and static media. Results. Differences in the Ly alpha/H alpha and Ly beta/H alpha luminosity ratios between the beta = 0.1 and 1 cases are factors similar to 2 for shock velocities 1000 less than or similar to upsilon(s) less than or similar to 4000 km s(-1), thereby providing a direct and unique way to measure beta. In principle, broad, " non-radiative" Ly alpha from SNRs in young galaxies can be directly observed in the optical range of wavelengths. However, by taking the different rates between core collapse and thermonuclear supernovae into consideration, as well as the duration we expect to observe such Lya emission from SNRs, we expect their contribution to the total Lya luminosity from z similar to 3 to 5 galaxies to be negligibly small (similar to 0.001%), compared to the radiative shock mechanism described by Shull & Silk ( 1979). Although broad, non- thermal Lya emission has never been observed, these photons are produced in SNRs. Hence, the non- radiative Lya luminosity is a part of the intrinsic Lya spectrum of young galaxies.