X-ray and infrared surveys provide efficient, and to some degree complementary, means of detecting and characterising active galactic nuclei (AGNs), with the infrared also providing an important probe of the host galaxies. To this end we combine the deepest X-ray survey from the Chandra Deep Field-South (CDF-S) ‘7-Ms’ survey with the deepest mid-infrared (5.6 μm) image from the JWST/MIRI Deep Imaging Survey (MIDIS) in the Hubble Ultra-Deep Field (HUDF) to study the infrared counterparts and point-source emission of 31 X-ray sources with a median, intrinsic, rest-frame X-ray luminosity of log₁₀(L_Xc^{0.5−7 keV}) = 42.04 ± 0.22 erg s⁻¹. The sample includes 24 AGNs with a redshift range, set by the X-ray detectability, of z ≃ 0.5−3, with the bulk of the sources lying at z ≃ 1−2, i.e. around the epoch of cosmic noon. Through a multi-wavelength morphological decomposition, employing three separate classifications (visual, parametric, and non-parametric), we separate (where present) the luminosity of the point-like AGN component from the remainder of the host-galaxy emission. The unprecedented mid-infrared sensitivity and imaging resolution of MIRI allows, in many cases, for the direct characterisation of point-like (i.e. unresolved) components in the galaxies’ emission. We establish a broad agreement between the three morphological classifications. At least 70% of the X-ray sources, including some classified as galaxies, show unresolved emission in the MIRI images, with the unresolved-to-total flux fraction at rest-frame 2 μm ranging from ∼0.2 to ∼0.9. At high X-ray luminosities (log₁₀(L_Xc) > 43 erg s⁻¹), we derive a consistent rest-frame near-infrared 2 μm point-source luminosity to that derived for local AGNs, whilst at lower X-ray luminosities we identify an excess in the 2 μm emission compared to pre-JWST studies. We speculate that this offset may be driven by a combination of Compton-thick AGN components and nuclear starburst, merger-driven activity. Our observations highlight the complex nature of X-ray sources in the distant Universe and demonstrate the power of JWST/MIRI in quantifying their nuclear infrared emission.