The game creates a supply depot for any detached land mass. The supply system (usually via convoys) will then dump supplies and fuel into that depot, and the entire connected landmass will then draw from that depot in the same manner that your core regions draw from your capital. As long as there are supplies left in the depot, and the units can trace an unobstructed line to that depot, the units on that landmass can remain in supply. Whether they get ENOUGH supply from that depot depends on techs and the infrastructure between them and that depot, as well as the amount of supply in the depot at the time it's cut off. Unfortunately, the local manufacturers do not add their supply and fuel outputs to the local depot, but magically teleport them to the capital, from whence they may be shipped all the way back to the depot in or near the city where they were produced. Any supplies or fuel injected into that landmass via transport planes, convoys, or other means will either be used by units in the province where it arrives, or else will travel overland all the way to the regional depot, to be distributed.
If you tag-switch to the UK, you can check any unit in India to see where it's drawing supplies from. There's the depot. Take that, or cut off access to it, and the troops will be down to whatever's already in the transport chain, plus their 15 day local supply.
Playing Japan, your supply depot is likely to be in Northern China, which makes operations all the way in India extremely difficult unless you go way out of your way to avoid (or lose) any land connection between that Chinese theater and your operations in India. All of that supply has to go overland through China, through the jungles and mountains of SE Asia, and eventually all the way to India. Good luck getting more than token amounts of food and ammo at the end of all that. Once the connection is broken, it will create a separate supply depot for the isolated area, but re-connecting the regions will merge the depots again, and it's unclear as to how it chooses which one to delete and which to keep.
Supply in HOI3 isn't totally broken, but it's got more than its fair share of quirks and limitations.