Really I think the answer is hold onto the American colonies, so you're looking at a POD in the early 1700s or possibly even sometime in the lateish 1600s. The fundamental issue the British Empire had was it was a tiny Metropolitan, with the vast and very distinct set of colonies while e.g. in Rome's case while it's true ultimately the population of the Empire dwarfed Rome/Italy each individual conquest tended to be smaller than the already romanised whole.
So have Britain hold onto its American territory and ideally have something that pushes greater centralisation instead of a Dominion-esque model and even if British North America doesn't expand that much (and I'd put pretty good odds on it taking at least the bulk of the Louisiana purchase although quite possibly not managing much more than that/still smaller than the USA), you have a big enough core population of people who predominantly identify as British, and a big enough Metropolitan economy that absorbing more parts of the periphery become more practical.
I agree with what others have said that even in this scenario I can't see them successfully fully integrating India is just too big but in this case I could see a scenario where alongside a fully integrated North America, Australia (if they go there in this timeline), and New Zealand you have a bunch of small island territories (Malta, Cyprus), and for that matter hanging on to a decent chunk of African territory (e.g. Nigeria or South Africa again if they ended up in the same areas as OTL). At an extreme stretch if you wanted the most powerful Imperial union possible I could even see a situation where India remains under a sort of formal protectorate agreement probably splintered into multiple states (likely successors to the princely states) and economically and militarily bound to the greater Imperial whole (alongside, Imperial bases on Indian soil and possibly even some treaty ports/strategic islands directly under Imperial control), but can't see a realistic scenario where large swathes of the subcontinent would be under direct control by an ATL modern day
So how to go about this I suspect if Britain is still in control of all of British North America, about the 1840- 1850s without having had to take a massively devolved approach, you'll see close union between the culturally British parts of the Atlantic become more and more the default as at this point travel to Parliament gets increasingly simple, and the telegram, makes communication between the distant parts of empire fairly possible. This does however mean going against the trend of British imperialism of the period. I wonder if perhaps the answer might be giving Britain more of a external threat after all the American colonies tended to be most loyal OTL when they were worried about France, perhaps a more entrenched and organised Franco-Spanish alliance, with the colonies in America worried about their colonial neighbours while the British home islands still eyeing another Spanish Armada. Also once you achieve that first step of home islands Britain comfortably absorbing its American colonies into the core part of the nation, that then becomes an accepted/acknowledged policy option when dealing with colonies with other problems if the circumstances are right helping to drive integration elsewhere going forward.
The other big hurdle here given the timing is how to get beyond a whites only federation given the rise of scientific racism. While it suggests a very very unpleasant world throughout the early 1900s suspect the answer is don't bother. Instead the initial federalisation is integrating African colonies with a substantial ruling white majority where whites are still the only electors (so places like OTL South Africa or kenya). Then have a World War II equivalent against a power that completely discredits eugenics followed by a slow period of democratising the black majority states of the Empire (so something more like the American civil rights movement) which steadily sees black citizens granted the vote and then in turn taking power in their home states but still part of the Imperial whole but at this point with the Empire both sufficiently powerful and sufficient unified that independence isn't really seen as an option.
