India (mined in present day India and said to be originally owned by Hindu rulers in the region)
Turkey (the Delhi sultanate - Tughlaq said to have taken it. Turkic in origin and was passed through the Delhi Sultanate)
Pakistan (the moghul connection - Babur was the first recognised owner. Lahore was its last place of residence before it was taken by the Brits)
Uzbekistan (the moghul connection - they originated from there)
UK (current owners)
Iran (Nadir Shah took it from the Moghuls)
Afghanistan (I think Shah Shuja took it from the Iranians)
The Sikh community (was given to them by the Afghans and they were last in possession before the Brits albeit in Lahore)
Does India even want it back? Didn't Nehru say "Diamonds are for the Emperors and India does not need Emperors."