The OP never stated any parameters, so definitive statements are somewhat pointless.
I think that even under the best of circumstances, true olitical union of the whole (or even the greater bulk) of the empire at its height would be unstainable in the long term. To keep it acceptable to Britain itself, Britain (or at least the "white" parts of the Empire) would need to have a majority vote forever, and the non-white parts of the Empire wouldn't accept that in perpetuity. At some point, it would be blatantly obvious that they have by far the greater population, and they'd demand proportional representation. This would indeed create an Indian Empire. Britain would not accept this. A split would be the outcome.
On the other hand, how much political union is required here? The OP, again, gives no details or clues. I can quite easily imagine a British Empire that consists of England, Scotland, (united?) Ireland, (maybe not-united?) Canada, Australia and New Zealand. Each of the constituent realms would have a Parliament of its own, and there would be an Imperial Parliament in London.
These realms would form a true Empire, sharing the same fundamental laws, one military (more especially: one navy) and exercising a common foreign policy. They'd obviously have a common economic space, and free internal migration with no barriers (as they'd have a common nationality as well, same as the states of the USA).
It's even possible to devise a scenario wherein this kind of thing is first set up as a solution to the grievances of the Thirteen Colonies, thus keeping them on board-- although that would presumably abort OTL's British interest in Australia and New Zealand.
Anyway, beyond these core parts, most other parts of the Empire (at its height) would be devolved into Commonwealth states. They'd still share the same monarch, but they'd be far less integrated. Basically soveign nations that happen to be in personal dynastic union, and which have close economic and military/defensive ties to the 'Empire proper'.
A few smaller odds and ends (Bermuda, the Falklands etc.) could be politically governed directly from London, either as special territories or as part of England(-across-the-Sea).
Potentially, South Africa could end up partitioned, with a rump state inhabited by Anglos, Boers and Coloureds remaining part of the Empire, and the rest of Southern Africa either becoming a set of Commonwealth states, or acrimoniously splitting away completely.
To my mind, this basic set-up absolutely qualifies as a "surviving British Empire". Even if India (by far the biggest player) opts out entirely and doesn't even join the ATL Commonwealth, the remaining Empire+Commonwealth is still a vast political system bound together (to varying degrees) under the auspices of one dynasty.
That actually sounds rather romantic. I suppose that such a British Empire could be close allies with both the US and India, right? I wonder if direct government rule will also apply to Malta, Gibraltar, and Singapore. And will Malaya (Malaysia) stay or become independent?