Partition is the experience being described. I would say that one of the best works to actually read on this topic would be Salman Rushdie's "Midnight's Children," which recounts the challenges in Partition. Essentially, when the British began to realize that their hold on India was slipping, it was involved in a problem. After occupying the nation for so long, it had done a good job in repressing the individual identities and cultural heterogeneity that had defined the people of the Indian subcontinent. They realized that the could not simply "leave" and expect "India" to the be the same nation that they had governed for so long. At the same time, Indian politicians began to recognize that as the British left, a void of leadership was created into which they could easily enter. The politics of declaring and fighting for independence for India became vastly different in ruling and controlling it. In the end, typical political horse trading ended up creating a Muslim majority in Pakistan and Bangladesh with the predominant Hindu majority in the new India. Certainly, as this solution might have been presented in a simple manner, it was far from simple as the exact social reality of Partition took a great toll on the new nations being formed.
I think that the word that you are most likely looking for is "partition." This word just means the division of some space into separate areas. You can have the partition of a room into two smaller rooms, for example. In this case, the term refers to splitting a country up into smaller countries.
As you say, there have been a couple different partitions in what was once British India. In 1947 Pakistan was partitioned off from India. In the 1970s, Pakistan was partitioned and became Pakistan and Bangladesh (which was created out of what had been East Pakistan).