The way I'm thinking about it is that you need the Empire to be something that India wants to be in. One key point would be having it as functionally a major global alliance network built on top of the British Empire itself - much like NATO or WARPAC in OTL - and in this case that probably means that China, Russia or both are hostile to the alliance in general and India specifically. (That means there's a reason for India to stay within the network, in this case to stay under the Empire's nuclear umbrella or the like.)
Actually, going back very far if Britain does better in the American war of independence you could see the emergence of an Atlantic straddling country governed similarly to the U.S. today.
Both of these can quite easily be woven together - but this rolls back the 'Empire' right to the earliest days. A resolution to the American colonies that actually... resolve issues and keep them in the Empire sets a totally different precedent (and flavour to Empire).
But an Empire with America and India- both of them benefit from the economic relationship of not having tariffs between them.
But sadly this goes back to the "We need to rebuild the Empire from the ground up", rather than make changes.
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An alternative to my hyper-liberal happy answer is the Strategic Division Council (or something like that), which is slightly more "in" than Saphs approach.
I.e. Dominion status for India, Canada, Australia-New Zealand (yes, together), and South Africa/Rhodesia/etc.
They, alongside Britain have a representative 'First Minister' or Dominion Rep. Each responsible for different spheres (which has a whole mess of repercussions strategically). Britain is responsible for European & N.Atlantic Affairs, Australia for Pacific, India for the Indian Ocean, and South Africa for S.Atlantic and African Affairs, Canada for Arctic & N.American Affairs. Now you could always argue more votes for those with more 'Concerns' - but it is explicitly NOT democratically handled, but strategic (almost a Stratocracy for Imperial Affairs, Democracy in the Dominions). So each rep has offices, and responsibilities for their 'Concern' but also are supported by the Military of their Dominion. So India would be the home of the Indian Concern Fleet. Britain the European/Atlantic, etc.
This means that India has its own government more or less, a vote on the joint decisions of all Imperial-level affairs, as does Greater South Africa, Australiasia, Canada and Britain.
Now, this does mean surrendering control over most of the daily affairs of India, but I could see it as a plausible alternative to independence post WW1/WW2.
Plus, besides some limited common affairs (foreign policy, cross-concern projects), the council is able to reform - India pushes for war in Afghanistan - who is in charge of that? The others are probably going to vote against India getting it, and instead setting up a 'Central Asian Concern'.
Now excuse me whilst I vomit at the idea of a British-Empire turned world-stratocracy.