Guys
Quite a challenge but not totally impossible. The idea path for me might be:
a) A shorter, more successful WWI. Say a small check for the RN at Dogger Bank and a properly organised Gallipoli. Those result in a shorter war, say ending in 1916 or early 1917, with a more successful Jutland resulting and a far more confident Britain still with much of its economic power. Its had a hard fight, which has shown the weaknesses of the old laisse-faire viewpoint and will prompt social changes. It also means changes in the post-war settlement to, to modify the old NATO saying, "keep the US out, Russia in and Germany down". Basically the US stays in isolation, Russia under either a weakened monarchy or moderate republic stays part of the political family, ideally Austria-Hungary survives in reduced form and Germany is split up, divide and rule to alienate the Prussian conservatives/Junkers from the bulk of the country.
b) With weaknesses shown by the conflict a major programme of social and economic reform, updating industry, greatly improving education and social conditions to strengthen the core of the empire. At the same time the ties strengthened by the conflict and concerns about vulnerabilities elsewhere a more serious attempt at Imperial Federation is made and has some progress. Since Britain is willing to be more interventionist, including tariff reform in the face of continued protectionism elsewhere, it is more likely to be able to win over the dominions. At the same time steps are taken to encourage reform in India and move towards self-rule.
c) Maintain the alliance with Japan and possibly also a period of naval tension with the US. This achieves a number of aims. If the US completes the historical 1016 programme it will give a number of bonuses.
i) The US will be left with an unbalanced and expensive fleet which will probably generate enough frustration in Congress that new construction will be greatly delayed.
ii) It will still be enough of a threat to Japan that, especially after the 1923 quake, that country will be more dependent on the alliance with Britain, enabling us to exert more influence and coupled with the stronger world economy very likely prevent the militant take-over.
iii) With a lot more money, still a lot of self-confidence and an improving economy the RN will not suffer the disastrous Washington Treaty with all the damage to the fleet and construction industry. Therefore construction will continue at a suitable rate avoiding the block-obsolesce Britain faced in the 1930s.
iv) If the navalists are loud enough in the US it might make the Canadians more receptive still to ideas of imperial federation, or at least co-operation on areas of common defence.
Those 2 steps set the pattern for a much stronger Britain and a larger state centred on it. Depends on the circumstances as to how things go. Should largely butterfly the Great Depression or at least moderate it. Even if, by some virtually ASB changes we get something like the world in OTL 1930s with an hostile Japan and reunited, expansionist Germany then Britain is in a much, much stronger position verses both those opponents while keeping Russia as a non-pariah will mean that any attempt by Germany to force drastic changes in Europe will be very short lived. As such any WWII equivalent conflicts should be fairly short and for Britain inexpensive.
In the longer terms I would see much of the empire move towards independence. Including India and most of the other African and Asian colonies. Ideally under friendly terms and with markedly greater stability and wealth. The awkward thing here might be the colonies in southern and eastern Africa with their white minority population. That gives the historically awkward problem of transition from minority rule to democracy. Ideally they might be included in the Federation on full terms and provide a good degree of stability to Africa as well as increasing the Federation's resources - population, raw materials, land etc significantly. Would also like to try and keep in the Federation at least some of the key strategic colonies - for instance Malta, Singapore, Hong Kong etc. Especially since some of those would also be important economic centres.
Given a multi-polar world with several major powers, the Federation, the US, Russia, possibly France and later Germany, China and India it should be possible to avoid a major cold-war period. There will be period of tension, especially with changes in the balance of power as non-European powers develop or at least gain independence. The Federation would probably have limited rivalry with the US, Russia, later China [over Hong Kong]. Enough to maintain a degree of unity in the face of external threats and decent interest in international affairs but not enough to led to serious conflict or social or economic strain.
The Federation, if at its smallest size, i.e. Britain, Canada, Australia, New Zealand and some of the smaller but sell placed locations would have a population of say 120-150M which would be enough, as a developed state to be a major power. Whether that would be a superpower would probably be a matter of definition, along with probably dependent on who else is about.
If larger, say successfully including the African lands, then its a significantly bigger power with greater capacity. If really successful it would include India and the Asian colonies and would be more powerful still. Under those circumstances it would probably be in danger of generating a potential hostile alliance of a number of other powers to balance it.
Steve