If the rebels manage to take the capital, the country's government will be replaced and all foreign relations will be reset. If you have already have them sphered or have a lot invested in them, that's a BAD thing, and you'll want to send a few brigades to put down the rebellion, if possible. I've done that in a number of spherelings, after requesting military access.
On the other hand, when a GP has major revolts worldwide (after a couple of wars against Russia and China in rapid succession) and its capital is taken by rebels, it's potentially a golden opportunity for everyone else, as all of their spherelings are released and their foreign influence is removed from all countries. The UK suffered a successful rebellion in one game I recall, and they lost all influence and their sphere in India, as well as everywhere else. It's not every day that you get to see a Belgian controlled India, or a UK that's near the bottom of the GP list.