We use the latest results from Atacama Large Millimetre/submillimetre Array (ALMA) surveys targeting the ionized carbon [C ii] 158 μm and oxygen [O iii] 88 μm lines, in combination with data-driven predictions for the evolution of neutral hydrogen (H i), to illustrate the prospects for intensity mapping cross-correlations between 21 cm and submillimetre surveys over z ∼ 5 – 7. We work with a data set including the ALPINE and REBELS surveys for [C ii] over z ∼ 4.5 – 7, and ALMA [O iii] detections over z ∼ 6 – 9. The resultant evolution of the [C ii] luminosity–halo mass relation is well described by a double power law at high redshifts, with the best-fitting parameters in good agreement with the results of simulations. The data favour secure detections of the autopower spectrum of [C ii] at all redshifts with an enhanced Fred Young Submillimetre Telescope (FYST)-like configuration. Such an experiment, along with the Murchison Widefield Array (MWA) will be able to measure the 21 cm - [C ii] cross-correlation power with a signal-to-noise ratio of a few tens to a few hundreds. We find that a balloon-borne experiment improving upon the Experiment for Cryogenic Large-Aperture Intensity Mapping (EXCLAIM) should be able to detect the 21 cm - [O iii] cross-correlation with the MWA and the Square Kilometre Array (SKA)-LOW out to z ∼ 7. Our results have implications for constraining the evolution of luminous sources during the mid-to-end stages of reionization.