Fast and massive winds are ubiquitously observed in the UV and X-ray spectra of active galactic nuclei (AGNs) and other accretion-powered sources. Several theoretical and observational pieces of evidence suggest they are launched at accretion disc scales, carrying significant mass and angular momentum. Thanks to such high-energy output, they may play an important role in transferring the energy released by accretion to the surrounding environment. In the case of AGNs, this process can help to set the so-called co-evolution between an AGN and its host galaxy, which mutually regulates their growth across cosmic time. To precisely assess the effective role of UV and X-ray winds at accretion disc scales, it is necessary to accurately measure their properties, including mass and energy rates. However, this is a challenging task, due to both the limited signal-to-noise ratio of available observations and the limitations of the models currently used in the spectral analysis. We aim to maximise the scientific return of current and future observations by improving the theoretical modelling of these winds through our Winds in the Ionised Nuclear Environment (WINE) model. WINE is a spectroscopic model specifically designed for disc winds in AGNs and compact accreting sources, which couples photoionisation and radiative transfer with special relativistic effects and a three-dimensional model of the emission profiles. We explore with WINE the main spectral features associated with the disc winds in AGNs, with a particular emphasis on the detectability of the wind emission in the total transmitted spectrum. We explore the impact of the wind ionisation, column density, velocity field, and geometry in shaping the emission profiles. We simulated observations with the X-ray microcalorimeter Resolve on board the recently launched XRISM satellite and the X-IFU on board the future Athena mission. This allows us to assess the capabilities of these telescopes in the study of disc winds in X-ray spectra of AGNs for the typical physical properties and exposure times of the sources included in the XRISM performance verification phase. The wind kinematic and geometry (together with the ionisation and column density) deeply affect both shape and strength of the wind spectral features. Thanks to this, both Resolve and, on a longer timescale X-IFU will be able to accurately constrain the main properties of disc winds over a broad range of ionisation, column densities, and covering factors. We also investigate the impact of the spectral energy distribution (SED) on the resulting appearance of the wind. Our findings reveal a dramatic difference in the gas opacity when using a soft, Narrow Line Seyfert 1-like SED compared to a canonical powerlaw SED with a spectral index of $