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Observations of the high redshift universe provide a promising avenue for constraining the nature of the dark matter (DM). This will be even more true with the advent of the James Webb Space Telescope (JWST). We run cosmological simulations of galaxy formation as part of the Effective Theory of Structure Formation (ETHOS) project to compare high redshift galaxies in Cold (CDM) and alternative DM models which have varying relativistic coupling and self-interaction strengths. The interacting DM scenarios produce a cutoff in the linear power spectrum on small-scales, followed by a series of "dark acoustic oscillations". We find that DM interactions suppress the abundance of galaxies below $M_\star \sim 10^8\,M_\odot$ for the models considered. The cutoff in the power spectrum delays structure formation relative to CDM. Objects in ETHOS that end up at the same final masses as their CDM counterparts are characterised by a more vigorous phase of early star formation. While galaxies with $M_\star \lesssim 10^6\,M_\odot$ make up more than 60 per cent of star formation in CDM at $z\approx 10$, they contribute only about half the star formation density in ETHOS. These differences diminish with decreasing redshift. We find that the effects of DM self-interactions are negligible compared to effects of relativistic coupling (i.e. the effective initial conditions for galaxy formation) in all properties of the galaxy population we examine. Finally, we show that the clustering strength of galaxies at high redshifts depends sensitively on DM physics, although these differences are manifest on scales that may be too small to be measurable by JWST.