Also note that both Pakistan and Bangladesh are Islamic nations. The only significant secessionist movements in India are in its non-Hindu fringes. Hindu India is united.
the violence was far less profound than at this point, of course some of that has to do with a big imperialist power occupying the country (the enemy of my enemy is my friend).
Hindu-Muslim violence has had its ups and downs since Islam first came to the Subcontinent in, I believe, the 8th century. You're right, Hindus and Muslims worked together to get rid of the British, but as soon as it was obvious the British were preparing to exit (well before the actual exit), Hindu-Muslim animosity spilled into the open and triggered a series of events that led to the emergence of Pakistan.
And even before the British came, there was significant Hindu-Muslim conflict, for example, during the reign of Aurangzeb of the Mughal dynasty. Bottom line, the religious conflict started the day Islam entered the Subcontinent and continues to this day. What we're seeing is just a continuation of a very old war.