Hmm, that's an interesting idea. Please continue.
OK.
Short synopsis is that the US stays out of WWI, resulting in it lasting much longer, and Germany also spends more building up and rebuilding the High Seas Fleet, forcing Britain to do the same in order to maintain its naval superiority. The combination of the two results in the war lasting until an armistice in January 1920. France is a mess across its northern half, and Britain is nearly bankrupt.
The Treaty of Versailles is much, much lighter than OTL, as the powers are less worried about Germany building back up owing to its financial situation, but Britain leaves the war almost insolvent. Facing that reality, Britain grants independence to Ireland in 1921 and decides to reorganize its colonies, with the aim of turning them into federations closely allied with Britain. This doesn't sit well with British Conservatives, but they can see Britain's financial situation and industrial unrest in the 1920s and realize the cost of keeping troops in numbers in the colonies is hurting the country's finances rather badly. Britain also realizes, however, that they could not grant independence to many of the colonies yet, owing to their problems with development. (Not that this stops many of them from trying in any case.)
The organization is a bit complex, but not that much so. Britain keeps immediate control of the United Kingdom, as well as Malta, Gibraltar, Bermuda, Cyprus, the Falklands and Palestine. The original plan included Palestine in a Middle East nation, but the independence of Britain's colonies in Egypt, Sudan and Iraq in the 1920s ends that idea and forces Palestine into direct control from the UK - a decision that will have consequences later. Canada takes over all of Britain's colonies in the Western Hemisphere save for the Falklands and Bermuda. Australia and New Zealand get the Pacific Islands, as well as Papua New Guinea and Hong Kong. South Africa gets South West Africa, Bechuanaland and Rhodesia, While the West and East African colonies are formed into federations as well.
India, however, is the biggest and most ambitious of the plans. British India had developed considerably further than the African colonies, and so London is more comfortable with India being the giant federation. All of British India is included in this, as well as Myanmar, Malaysia, Singapore, Ceylon, the Maldives, Aden, Oman, the United Arab Emirates, Socotra, British Somaliland, Mauritius and Britain's Indian Ocean Territories. This federation surprises everyone, largely because of its incredible size and population. India's leaders are split over it, but all see taking on that challenge as their ticket to independence from the United Kingdom. The Arabs at first are not impressed with the idea, but they come to see the logic of it.
Negotiations between New Delhi and London hammer out the agreements. Federal India will have to be a fairly loose federation, necessity demands. But as the Raj was primarily this way in any case, that didn't bother too many people. Negotiations are similar with all of the others.
In 1925, Australia formally reforms the Commonwealth, which turns the six-province federation into one with eleven provinces as New Zealand, Papua New Guinea, Fiji and Hong Kong join in. Britain pledges to assist the Australians in the job of integrating these areas, and do follow through on their promises. The following year, South Africa does the same, formally bringing in South West Africa, Bechuanaland and both Northern and Southern Rhodesia into South Africa. The Dominion of Canada becomes the Canadian Federation in 1928, doing the same as what South Africa and Australia had done.
After years of negotiations and diplomacy, the Indian Federation is finally formed in 1928 and becomes nominally independent the following year. India remains part of the British Empire, but is given wide control over its own affairs, and much of that is further still devolved to the many provinces. Britain stays in the East and West African colonies, preparing them to be independent themselves.
Over the 1930s, the Great Depression hits everyone, but everyone largely follows the ideas of Keynesian economics, which between better trade relations and extensive civil works projects helps India's economic situation. This coincides with Britain nearly going completely under due to its debts and the Great Depression, and as a result a large number of Brits move aboard, focusing on the areas which due to the federation agreements they can still go to. India in particular gains a fair bit of notoriety in this regard.