The British ruled over and administered India for the best part of 100 years.
They started trading with India in the early 1600′s. By the mid 1700′s they were engaged in fighting against various kingdoms. By the mid 1800′s they were fully in control of the Indian subcontinent and were either directly administering through the means of the Indian civil service, or were letting the local Maharajahs administer their own Kingdoms as long as the Maharajahs were paying the British the desired license fee and were prepared to follow the terms and conditions set out by the British.
In 1947 the British left India partitioned into India and Pakistan as a result of which over one million people died in migration between the two new countries. In addition India was virtually bankrupt with very little infrastructure
or industry. Yet when the British first went to India in the 1600′s it was a very prosperous place with a large GDP and a huge manufacturing base, which is the reason why the British wanted to trade in the first place. The conclusion being that the British left India in a far worse state then how they found it.
Since 1947 India has developed all its infrastructure and industries from ground up, including roads, agriculture, textiles, steel, nuclear, space, military, telecoms and IT and much more. It has made so much progress that some consider it to be a potential global superpower in the very near future, and this has all happened within 60 years of The British leaving India.
So what really was the legacy of the British in India? Many would say that the British went into a prosperous land, bled it dry, and then left. Others would say the British brought culture (education to the heathens), technology and of course the great English language to a country that otherwise would never have experienced all of this.
But one point that often gets missed is that India is now a united country with a democracy and all the relevant institutions needed to sustain democracy, like a judiciary and armed forces. The British had a big hand in this although their motives were all together different. Basically when the British came to India in the 1600′s, it was a fragmented country with many little kingdoms none of whom liked each other. The British took advantage of these differences and slowly gobbled up each kingdom, eventually bringing the sum total of all these kingdoms under British rule. Eventually when the British left they handed over a united India.
It seems that this unification could only have been achieved by an alien power like the British because the Kingdoms hated each other to the extent that they would join the British against each other, yet they would not join each other against the British. This character flaw in the Indian persona is exactly what the British exploited and is also why it would have been very difficult for the Indians to have unified by themselves without some external influence.
India has made great progress over the last 60 years because it is unified. Surely this has to be counted as an important British legacy.