While nations land-locked by a single other nation do exist, they are very few enclaved countries and they only exist due to unique circumstances. Vatican city and San Marino are examples where Italy could not, or would not absorb these countries into it. India on the other hand, has no such qualms. There is no reason for an independent state of hyderabad to exist if a united (or even semi-united) India exists.
Setting aside the nationalism issue, since that would require innumerable PODs, we can try to see how such a rich and big nation could defend itself from the bigger nation surrounding it. Hyderabad is not surrounded by mountains since it is on the deccan plains and IRL the Indian armed forces managed to defeat and annex the state of Hyderabad within a period of 5 days. I do not see the annexation failing, even if it requires a year or two since Hyderabad is part of the deccan heartlands, has no other borders than its borders with India and is pretty much indefensible.
Someone mentioned that Hyderabad would buy Goa but the Portuguese kept it as a colony for prestige, not money. India had to invade it and kick out the Portuguese and I do not see Hyderabad doing anything different.