  We use the latest results from Atacama Large Millimetre/submillimetre Array
(ALMA) surveys targeting the ionized carbon [CII] 158 $\mu$m and oxygen [OIII]
88 $\mu$m lines, in combination with data-driven predictions for the evolution
of neutral hydrogen (HI), to illustrate the prospects for intensity mapping
cross-correlations between 21 cm and submillimetre surveys over $z \sim 5-7$.
We work with a dataset including the ALPINE and REBELS surveys for [CII] over
$z \sim 4.5-7$, and ALMA [OIII] detections over $z \sim 6-9$. The resultant
evolution of the [CII] 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 auto-power spectrum of [CII] at all redshifts with an enhanced Fred Young
Submillimetre Telescope (FYST)-like configuration. Such an experiment, along
with the Murchinson Widefield Array (MWA) will be able to measure the 21 cm -
[CII] 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 - [OIII] cross-correlation with the MWA and the Square
Kilometre Array (SKA)-LOW out to $z \sim 7$. Our results have implications for
constraining the evolution of luminous sources during the mid-to-end stages of
reionization.
