The later the project happens, or the longer it it is sustained
...the more that the settler dominions will have a greater say in how the federation is structured and operates, thanks to growing population, economy, and a greater political maturity. They'll demand it. And quite reasonably so.
1. The most common proposal I've seen was a true imperial parliament, representation largely proportional to population. Its power would be mainly confined to defence, foreign policy, customs and trade; dominions would retain their own governments to handle everything else. It was even proposed that this might be extended to the British Isles as well, with England, Scotland, Wales, and Ireland (or parts thereof) gaining their own parliaments for internal governance. In this situation, then, London would presumably host both the imperial and English parliaments, albeit likely in different buildings. While the British PM would likely wear both hats in the early years, this likely wouldn't last for more than a generation or two. MPs might be elected or appointed, but in the long run, legitimacy and popular support will surely demand popular election.
2. Imperial Parliaments would meet in London, surely, but as transportation became swifter and more reliable and the dominions grew in power, there would be pressure for at least token sessions to be regularly held in the Dominions, too.
3. One imagines that if it wasn't explicit at first, there would soon be a demand that every dominion region always have some minimum of presence on the imperial cabinet. Eventually, this would likely extend to major military commands as well. Judging by Canadian recalcitrance during the great Naval Scare of 1909, it seems likely that naval construction would be increasingly parceled out to new shipyards in the Dominions, though one imagines that only Canada would be in position for construction of capital ships for a fair bit. Likewise, force deployments and military bases will increasingly reflect dominion interests - Aussies and Kiwis will certainly extract major permanent RN fleet commands in the Far East/South Pacific, and RAF air commands would eventually follow suit.
4. Likewise, there will be more in the way of local communications and transportation infrastructure.
5. But sensitive issues would remain, and they would be difficult to address. a) Would membership be extended to any other colonies, nearly all of which would be white minority, and under what rules would such determinations be made? What about South Africa? b) Would regional associations and structures be permitted or necessary, i.e., some kind of local congresses between Australian or Canadian dominions to address purely regional issues of development? c) How would immigration policies be sorted out once that became an issue (as it almost inevitably would) in later 20th century? d) How would a federation handle decolonization, however it developed in this very different timeline - especially of the Raj? (Certainly there would be greater motive to keep some kind of control over the Suez Canal.) e) How eager would a federation be to dismantle any imperial preference and enter into trade deals with other major powers, especially America? f) How would local territories be governed and developed, i.a., Yukon, Northwest Territories, Australia's Northern Territory?