British Federation

Well, for one thing, the Dominions (Canada, Australia, etc) are already separate Crowns and the title King of Canada, King of Australia and so on is already a separate title from the UK crown. I'm not sure how a union of the crowns could be effected.

Most of the Dominions woud balk at the idea of being run by an overseas legislative body, even if they had representation.

Historically speaking, Canada has had a tough time maintaining federalism just in Canada alone, I cannot imagine how much more difficult things would be to maintain a federalism so much larger!

Oh ... well ... none of them have internal constitutions in 1923.

Foreign and military powers would probably be very very difficult after WW1, particularly in Canada's case (Aus etc might be easier),  treasury powers would be difficult at any time.

If you pushed things back 10 years foreign and military powers would be alot easier, but would likely have a detrimental effect on the battlefield in WW1. One of the main reasons Canadian forces were so effective in that conflict is because the Canadian Corps and Arthur Currie were able to operate relatively independantly, and they were very innovative.

Building on its reputation and effectiveness during that conflict, Canada demanded (and received) sovereign control of foreign and military affairs.

Keep in mind that the Commonwealth was considered a superpower right up until WW2 (in fact, the term superpower arose as a descriptor for the Commonwealth during the interwar period). I can imagine some AH scenarios where the Commonwealth remains a superpower after WW2 - mostly involving a (temporary) peace accord with Germany after the fall of France, and not signing on to the Atlantic Charter or Bretton Woods but promoting the Sterling Area and the pound as global reserve currency. Instead of the Bretton Woods currency regime, the Sterling Area is promoted in Africa, Asia, and perhaps Latin America eventually. This was do-able barring British economic devastation from the war.

Well ... essentially it would take a role not unlike the EU does today. A tentative partnership with the US where interests coincided (which would not be uncommon), but rivalry and competition where they differed.

For my part, it's mostly because the most important of the White Dominions - Canada - had little or no enthusiasm, after WW1, to have its foreign affairs dictated from London, let alone having its military commanded by non-Canadians any longer. It had dropped the pound and was no longer part of the Stirling Area, either, so it's obvious that economic interest in a greater union was a no-go as well.

Australia was actually pushed into independance kicking and screaming, so the "Federation" might have kept Aus and NZ but that's not really enough to prevent it from being drawn into the US orbit as a satellite. Ireland obviously could not be kept, and South Africa would have been a drain just like India (not to mention a huge political liability).

Some things in history turn on freak chance, like the Bolshevik coup during the Russian revolution. The Whites might have retaken the country, or the Mensheviks might have prevailed. But other events in history are long-term developments that don't depend on any single little variable or man-made decision, but are an inevitable product of the course of history; resisting them is like resisting a force of nature, it can be done, but the costs and lack of benefits eventually make it ludicrous to go on. The devolution of power to the Dominions is one such event.