Is the phrase "the subcontinent" well understood?
Solution 1:
This may vary depending on where you are and whom you’re talking to. In most of the communities I’ve lived in (England, northeastern US, eastern Canada), it was certainly used and understood; I think it’s fairly widely known.
However, if you’re finding some people are puzzled by it, you could instead say the Indian subcontinent — this is an almost-as-common version of the phrase, and should hopefully be clear to anyone.
Solution 2:
As someone living in NZ I'd give a different answer to Teylyn - that is, I judge that in New Zealand the word 'Asian' used of a person or attribute without further qualification would in the first instance be though to refer to someone of Chinese/East Asian background and only secondarily to someone from the Subcontinent.
Certainly what I understand to be the British usage (as per Marcin), where 'Asian' refers in the first instance to people from the Subcontinent, is not so commonly heard in New Zealand.
As for 'subcontinent' itself, I use it often and it is usually understood to mean the Indian subcontinent; I wouldn't hesitate to add 'Indian' if need be . . but if talking to a Pakistani or Bangladeshi or in a context where reference to those countries is implied, would more likely use South Asia.
Solution 3:
People in southern Africa (roughly south of the equator) often refer to their part of the continent as "the subcontinent", so to avoid ambiguity you should speak of "the Indian subcontinent" if that's the one you mean.
Solution 4:
This isn't really a question about the English language: it's a question about how (some) people view the world. The Indian subcontinent is, factually, part of Asia: as such, you can use the term without being subject to penalty of law. That said, the British Isles are factually part of the European continent, but I know plenty of people who would object to calling England (or even Britain) an European country. So you need to consider your audience; if they consider "the subcontinent" a descriptive, non-derogatory term, then using it is helpful. If not, you should not use it, whatever your personal views.
This seems to me to be a perfect example of a problem with stackexchange's obsession with objective answers, namely that the correct answer depends on the intended audience. The 'correct' answer to a question somebody asks you in a bar would not be the same answer you'd give in a witness box, even if the question was identical. Linguists refer to this as 'different registers'; there doesn't seem to be a solution (or even a description) in ordinary life.