The characterisation of the atmosphere of exoplanets is one of the main goals of exoplanet science in the coming decades. We investigate the detectability of atmospheric spectral features of Earth-like planets in the habitable zone (HZ) around M dwarfs with the future James Webb Space Telescope (JWST). We used a coupled 1D climate-chemistry-model to simulate the influence of a range of observed and modelled M-dwarf spectra on Earth-like planets. The simulated atmospheres served as input for the calculation of the transmission spectra of the hypothetical planets, using a line-by-line spectral radiative transfer model. To investigate the spectroscopic detectability of absorption bands with JWST we further developed a signal- to-noise ratio (S/N) model and applied it to our transmission spectra. High abundances of methane (CH_4_) and water (H_2_O) in the atmosphere of Earth-like planets around mid to late M dwarfs increase the detectability of the corresponding spectral features compared to early M-dwarf planets. Increased temperatures in the middle atmosphere of mid- to late-type M-dwarf planets expand the atmosphere and further increase the detectability of absorption bands. To detect CH_4_, H_2_O, and carbon dioxide (CO_2_) in the atmosphere of an Earth-like planet around a mid to late M dwarf observing only one transit with JWST could be enough up to a distance of 4pc and less than ten transits up to a distance of 10pc. As a consequence of saturation limits of JWST and less pronounced absorption bands, the detection of spectral features of hypothetical Earth-like planets around most early M dwarfs would require more than ten transits. We identify 276 existing M dwarfs (including GJ 1132, TRAPPIST-1, GJ 1214, and LHS 1140) around which atmospheric absorption features of hypothetical Earth-like planets could be detected by co-adding just a few transits. The TESS satellite will likely find new transiting terrestrial planets within 15pc from the Earth. We show that using transmission spectroscopy, JWST could provide enough precision to be able to partly characterise the atmosphere of TESS findings with an Earth-like composition around mid to late M dwarfs.