Biggest problem is that the two cores (the US and the UK) do not want to cede any power to the other. Plus, with one being heavily republican and the other monarchist, and the one isolationist and the other interventionist, combined with a myriad of other issues, and this isn't an easy problem.
Some sort of Imperial Federation incorporates all but the US, but it misses the intentions. The other part of the problem is trying to incorporate two different aspects. The US alone would feel rather isolated in the federation.
I'd propose a method that, on one hand, depends on a reduced British Federation (which, granted, would include all the settler Dominions, the Caribbean, South East Asia, etc) that is missing India and some other vestigial regions on the edges. on the other side, have the US maintain better relations with the Philippines (a greater investment in the Philippines, even if just in men and personnel rather than pure infrastructure. The early Filipino politics was an extremely local affair). Essentially, a statehood party that succeeds more and eventually morphs into a pro-US alliance party. Combine that with Cuba remaining a US protectorate.
Then, have the US sphere morph into a supranational union that results in some outsourcing but also investment into those US aligned nations. (this could include the Latin American countries). And, for whatever reason is appropriate in the timeline (Russia, Germany, etc), the USTO and the ComNat drift together just because they secure regions that concern both of them (the Pacific and the Caribbean) while providing the advantages of a common tongue and history.
Perhaps, once South Africa eventually goes to apartheid (murdering butterflies here), there would be a greater reaction by the ComNat, which could involve into a combined action by both sides? The South Africa intervention eventually becomes the defining moment where we have the creation of a supra-supranational organization, the beginning of economic integration and inter-organization movement being lowered.
Obviously too simplified, but of the two organizations, the Imperial one has the UK, Canada, Australia, the Caribbean Federation/etc, Malay States/Sarawak/Straight Settlements/North Borneo, etc, while the Republican one is the US, South Africa, Ireland, Cuba/Latin America, the Philippines, etc. (South Africa really drifting later on, while still technically ComNat)
Two roughly equivalent sides, with the "core" UK (UK/Canada/Australia/New Zealand) as a whole is, while maybe not as strong as the US, is close enough that there is parity. The UK might even retain the Straight Settlements, many of the small islands, etc. Hong Kong would be... interesting). The US sphere, while smaller and top heavy, is roughly equivalent and, in the end, it is an alliance between two equal alliances that grows.
Granted, the government might start resembling the HRE soon enough...