(For this question we'll define "imperial federation" as a federation of Britain and its settler colonies, give or take some islands and other small outlying territories)
The concept of an imperial federation is an idea that I often see tossed around on this forum, seen by many as a viable path for the British to maintain their influence and power around the globe. A cool idea, sure, but what incentives does it have to stay together in the long run as a federated entity united to any meaningful degree? The challenges I see are as follows:
- Economic. The constituent states (UK, CAN, SA, AUS/NZ) all have significantly different economic interests. Canada with the US, the UK with continental Europe, AUS/NZ with Asia, etc. For example, why should Canada allow trade with the US to be influenced by people thousands of miles away? No amount of integration is going to change the fact that each constituent state's largest trading interests lay outside of this hypothetical union.
- Cultural. I don't see how you can prevent the dominions from developing distinct, separate cultures and identities from the British -- not when there are thousands of miles of ocean between them. And when that happens, it's only a matter of time before voters begin to ask why those people get to have a say in their affairs and why those people get to have a share of their paid taxes.
- Military. The defense concerns of each state are wildly different. The threats that the British Isles face are not the same as the ones faced by say, Australia. They could probably afford a strong fleet, sure, but who's going to pay for it? Where does it focus its attention to? For example, if the US Navy was willing to keep shipping routes open and keep the waters safe, then what reason does Canada have to pay taxes towards the defense of Australia and New Zealand?
I want to believe that an imperial federation could survive well into the 21st century as a world power, as I do think it's a pretty cool concept. But the more I think about it the more impractical it seems. I'd love to be proven wrong though.
You have laid out the challenges to such a state pretty well. These would all be issues that would have to be dealt with if the IF actually existed. But I don't think they are insurmountable. Lets look at them separately:
Economics
These states would indeed have different economic priorities. But it could also be said that the East Coast of the US often has wildly different economic priorities from the West Coast, and that both can have wildly different Economic priorities than the Mid-West. Having different, or even incompatible economic priorities does not
necessarily preclude being part of the same nation-state.
In the case of Canada, it is worth noting that the Canadian economy was massively influenced by the American one since before Confederation, yet American trade with Canada only surpassed British trade in the 1920's in the wake of WW1. This was pretty quickly reversed again when the US increased tariffs, and was further reinforced by Imperial Preference on the Commonwealth side and even higher tariffs on the American side. If the economic situation in the Post-WW2 world is better managed or if the IF as a whole is doing well enough to take up the slack it is entirely possible for Intra-IF trade to be more important to Canada than trade with the US for much longer. US trade is still going to be a huge part of the economy and I would expect Canadian voices to be the strongest ones in advocating more open borders with the US, but it does not necessarily mean they will reject the IF for the sake of US trade. Indeed, the IF as a whole will likely have a strong resource export sector and even if the Dominions are more industrialized than OTL and Britain's industrial sector is more robust the IF as a whole would likely benefit from closer trade ties to the US. So the two are not mutually exclusive.
In the case of Australia and New Zealand, much the same export materials that now fuel their economy through trade with China used to fuel their economy through trade with Britain. An IF that has a collective industrial capacity greater than OTL and open borders between its members would likely still be an important part of their trade. And, like with the US, a properly managed relationship with South East Asia could be a benefit to all members and may not be mutually exclusive with an IF. There may also be an opportunity depending on the IF's relationship with India. If a greater integration between Britain and the Dominions is possible it might also be possible to have a better managed relationship with India. Depending on the path TTL's India takes it could end up filling the OTL role of China, taking in large amounts of raw materials and making use of its cheap labour to produce finished goods. If both these things are true then Australian and New Zealand exports will benefit from the closer relationship.
I am not well versed in South Africa's economy but they also have a large resource export sector, so much of what is said about the others applies to them as well.
Culture
A few years ago I moved from the part of Canada that I grew up in to another part far from it. I have spent much of the time since then trying to adjust to the culture shock. Though we were part of the same country the culture, expressions, attitudes and outlook on life were very different. Our political views are on many things almost polar opposite as well. Yet there was still a sense of connectedness and belonging since we were all Canadians.
For an Imperial Federation to work, you would have to have something similar. An overarching cultural identity that says "Even though you are a South African, and you are an Australian and I am a Canadian and he is from London, we are all (British/Imperial/whatever name for the larger culture you come up with)". That is part of why I believe that the time for the establishment of the Imperial Federation would have had to have been pre-WW1. WW1 gave a lot of life to the nationalist sentiments in the Dominions and somewhat damaged the interconnectedness of the Empire as a whole. To make the IF work in the long term I think you would have to have it existing (even if somewhat nebulous) prior to the war. Preferably you would also manage to turn the war into an exercise in nation building the entire Federation as well. Where people come out of it feeling as or more proud to be part of this larger entity than they went in.
Its probably the most difficult challenge to creating a lasting IF, but I don't think it is impossible.
Military
AIUI a big factor in Australia and New Zealand turning from Britain to the US militarily was the simple fact that Britain could not safeguard their integrity and this had basically been demonstrated by WW2. If the IF as a whole is in a strong position coming out of WW2 then it should be pretty possible to maintain a military that is capable of both preparing for WW3 (the main concern of Britain and Canada) and fighting the little conflicts throughout the globe (the main concern of Australia and New Zealand). This is basically what the UK was trying to do IOTL, but did not have the resources or political will to maintain.
So, to sum up, there were very good reasons why the Imperial Federation did not happen IOTL. But being impossible to maintain is not, IMHO, one of them.
