I just hate the revisionist stuff which tries to paint the mutiny as the 'First Indian War of Independence'. Its just...cringe worthy. And putting modern thinking into events where they don't belong.
Britain did not rule India at the time. It was the mutiny which brought about British rule. India was a complicated place. Officially the mughal empire still ruled much of the continent but even at their peak they were rather HREsque with lots of local kings remaining powerful under them...and of course there being lots of local rulers who weren't under them at all. And then despite this state of affairs you also had the EIC ruling in parallel; by this point in time without a doubt the most powerful force in the area but still not the official rulers....and then you've the complicated situation of the EIC's links to the British government. The links were big for certain but they were different bodies.
So...the rebellion gains more success....then what?
Are they going to throw the EIC out of India entirely?- will they confiscate its property? Who will it go to? What about other British interests? Just reset the power balance so the Mughals become the most powerful entity?
How can they keep the British out of India? A lot of people's livlihoods depend on British money.
It was a messy, messy situation with too many factors at play for me to even attempt to come up with a proper answer. It really was not just a simple question of India fighting for independence.
I'm not sure that
any of the more substantial answers in this thread have tried to paint it as India seeking independence.
What it would be is a shift in the balance of power- you're right in that the EIC, in essence, operated as a power faction within the nominal Mughal Empire, ruling "on behalf of" the Mughals in Bengal and the Gangetic plain.
What would happen is the same thing that might happen in any loosely held together Empire when one faction falls from power. Local rulers take back their privileges (as was always the case in India whenever central control lapsed) and on a wider level, whichever new central power rose would take over. In effect this would mean that the EICs powers, property and privileges would devolve to whichever faction managed to gain the upper hand in the wider war. Depending on the POD there might be various candidates- perhaps an alliance of Rajputs and Mahrattas under the charismatic Rani of Jhansi, or perhaps the Khalsa becomes a dominant faction (if the POD is before the Sikh Wars). All this would be under the titular overlordship of the Padishah in Delhi.
In the South, the case is that Britain is very clearly no longer top dog in India. It'll be a massive loss of face in Europe too and one suspects the other European Powers will start nosing around India. What may happen is that states like Mysore and Travancore will be transformed into a lot of Thailands, so to speak, playing a balance of power between the European powers (just as they did in the 18th C).
British dominance in India was based on huge amounts of luck and was always a delicate balance. Once the balance is upset there's nothing much they can do to reinstate it.