That's not entirely fair to the British. They actually had a pretty firm hold on the subcontinent at first, and splitting it up made things easier to manage. It was only after the Great War that British control really disintegrated, and even that took time.
Of course, the British aren't going to be able to hold onto India for forever, but they could well maintain control of it for another few decades, into the middle of the 20th century.
As to the actual OP, I find it hard to see how it could remain one big territory. There are simply too many groups of people in India, and there is so much land and population to sort out. For a foreign power controlling them from afar, the only way to maintain control was to split it up. This also allowed the British to play off of local rivalries and keep the area relatively passive. Parliament would have to be given a heavy dose of stupid not to change things.
In a similar vein, there's no way this monster would stay together after it went independent. That would be like asking all of Europe to be ruled by Berlin, Paris, or Brussels- and Europe for the most part doesn't even have the religious problems India does. They could very well unite to drive the British out, but at best we're looking at a few slightly larger states.