With any exercise routine, the most important bit is the 'recovery' between sessions.
You do the exercises to influence the way your body operates all the time, not just to operate it differently during the exercises. That means you need to give your body time to adjust its operation, after you have supplied a 'stimulus', before giving it another.
In the case of Dr Mosley's experiment with HIIT, the exercise was really 'Total Intensity', rather than 'High Intensity'. He was literally working AS HARD AS POSSIBLE during the short bursts that he put in, which affects the amount of 'recovery' that you need.
Basically, the harder you work, the more recovery is needed before exercising again. He's medically trained, has access to excellent physiotherapy support, and he was doing the whole experiment under the supervision of other doctors. So he could 'push' the amount of recovery he included in his regime, because anything that did go wrong would be spotted, diagnosed and treated immediately.
In reality, if you are performing the HIIT session with similar maximum effort, you should probably allow at least two days of recovery between sessions, as well as an extra rest-day every three or five sessions.
Any additional exercise that you do between the sessions also requires 'recovery', and will add to the total number of 'rest days' that your body needs.
Cutting corners by reducing the amount of recovery time you take is a waste of training, especially with HIIT, because your maximum performance will be reduced if you try to train before you are fully recovered. This effect is 'cumulative', so that repeatedly 'overtraining' (under-resting) will gradually prevent you from working hard enough to produce a positive training effect.