The Centre of South Asian Studies has a large collection of oral history and other sound recordings. Many relate to the independence struggle, but some are also about Partition. You can query for Partition in the search bar.
This collection of files from the Foreign Office (later the Foreign and Commonwealth Office) and Dominions Office focuses on the political and social history of India, Pakistan and Afghanistan. It includes some documents on Partition. Narrowing your search by year or using the advanced search will help.
This project contains a collection of stories told by people who left Bengal after Independence in 1947 when the state was divided into West Bengal and East Pakistan (later Bangladesh).