First I would use a 1914 map of the British Empire as I believe the Great War is the first mortal blow to the Empire that needs to be avoided. Now I would argue that Irish home rule is the first thread that unravels the Empire, something that is beyond a post-1900 solution, but not without impact. The conquest of Ireland and its incorporation into the British Empire began the Empire but also poisoned the well, home rule might have been accomplished to stave off the shattering of the Empire and I think avoiding the war was the best way to keep Ireland in the British orbit while paving the way for an Empire that could survive. Still a very steep climb. The Great War alienated most of the Empire for the sacrifices made, the Australians walk away rather disgruntled and the Indians begin to feel they die for the British in their wars, avoiding these helps. But I think India was too different and possessed its own independent history that meant it was going to depart, but could it at least stay in a defensive and economic union? I think if India goes then much of the Empire becomes a burden but not entirely, as trade expanded in Asia the chain of outposts still have value and lesser pieces between traditional opponents could be cajoled into staying associated. So I think you can evolve to an Anglophone Sterling zone of the White dominions and some of the minor colonies, but it might not be the Empire headed from London, more a sort of international organization with delegates in London bickering, better trade links and a common valuation currency exchange pinned to Sterling, all but the least populated and most vulnerable are anything but independent nations, yet they all might accept a symbolic British monarch as head of state. So is this the Empire? No. Is it something more than where we are? Yes. That is the best that can be done as we look forward from the decade before the Great war and hope to achieve.
This is the British Empire I have conjured for a post-Great War where the UK was not a belligerent, it is weaker where the Empire fights and Germany is not quashed, it devolves more fully and it is more broken apart but still a recognizable "British-sphere" because the USA does not become the super power or by the "special relationship" subsume the British after dismantling the Empire, the USA is cordial but not allied, hostile to colonialism it really only cares if it really has interests at stake, its military power is that of a peer and more focused, etc., so the British are more independent like the French but actually more independent, still a great power but far from being the power they once were a player in a more balanced multi-polar world.