Archives and special collections hold many items relating to India in the papers of politicians and individuals. For example, the papers of several politicians including Neville Sanderson, Peter Shore, David Steel and Len Jeger.
There are a lot of papers and correspondence from people and societies on a range of issues throught the 20th century relating to India, for example, Sir Hugh Beaver, the Fabian Society, William Piercy and the anthropologist Charles Seligman.
Also from the Fabian Society are the papers from the Fabian International Bureau detailing the impact of immigration from India to Britain.
The Sir Gilbert Longdon archive covers a wide timescale from 1925-1992 and consists of personal correspondence and photos and a transcript of interview with Mahatma Gandhi in 1945.
Health and law are two areas with interesting collections. Relating to law the Courtney papers show Indian Criminal Code confessions from the late 19th century. There are several archives relating to health issues from Florence Nightingales papers on village sanitation in the late 19th century to the Abel-Smoth papers on health policy and insurance.
On more economic matters there are the John Kydd paper on the Indian tea industry in the early 1920s and the Cannan Collection which includes the Royal Commission on Indian Currency and Finance in 1926.
Note that primary source descriptions often use contemporary place and country names only, as they occur in the original documents, so bear this in mind when choosing search terms.