The BCCI can't play against themselves, and after Pakistan, the biggest money maker for them are Australia and England respectively. If England and Australia want something done badly enough, they can definitely force the BCCI's hand.
I am not sure if it's accurate to say that the ECB and the ACB need the BCCI more than the other way around. Perhaps individually, they do. But together, they make a very strong and compelling front. The 'problem', depending on your point of view are West Indies and South Africa.
If such a thing came up, BCCI would obviously gain support of the other Asian countries, but a worrying thing for the ECB/ACB is that their traditional supporter in West Indies is being pried away to vote with BCCI on most matters (big example was the WC rights). South Africa, to a much lesser degree also fall into this category, though since their re-admission they've always been very pro-BCCI (they played their first match against India) and generally vote with them.
Basically, the worst fear would be that England and Australia would be stranded alone with only New Zealand as the support, and that they cannot afford. On the other hand, even if the BCCI gets everyone but the ACB and the ECB, they too would lose a huge chunk of their income, and all sides would be much worse off than they are now. And if the worse case scenario for the BCCI comes up (they lose everyone but the Asian countries plus South Africa), that would pretty much destroy whatever money is going into the Indian domestic game, and make it even more terrible than it is now.
It is not really in anyone's interest to break apart. It's sort of like the UN. Far from perfect and a whole lot of petty bickering goes on, but the alternative of having no united organization to at least try to get along is much much worse, and for everyone.
I know certain people, such as Rich (and I think Neil too) think it may not be a bad idea if there was a split, but I firmly believe it would really destroy the international game or at the very least be a massive massive step backwards, and the effects of that would reach every level (down to the playground) eventually.