Numerical simulations were made of the interaction of a relativistically intense laser pulse with a target consisting of nanometre fibres. Fast electrons were shown to execute forced betatron oscillations in the electrostatic fibre field and the laser field. The fibre diameter was determined whereby the amplitude of betatron electron oscillations is resonantly increased. The power of coherent X-ray betatron radiation of the electron bunch was calculated outside of the resonance domain and in the resonance case. We showed that the laser-To-X-ray betatron radiation conversion coefficient in the resonance case amounts to a few percent and the target made up of nanometre fibres may be regarded as an efficient laser-driven source of coherent X-and gamma-ray radiation.