The real problem with this is economic.
An EU-like trade block with countries scattered around the globe is going to be difficult.
The EU is a compact body, most of the nations do most of their trade with each other, and it makes sense to set up a common market.
Canada, at least, trades and traded far more with the US than with Britain, and had minimal trade links with Australia and NZ, afaik. (OK, we got our lamb from NZ, oh, and kiwi fruit, but that's hardly enough to make a major trading block.)
Britain probably wouldn't mind a bloc where Oz and Nz sold food to the UK in exchange for manufactured goods - but both those countries presumably want to make their own goods. And if they don't, the fact is that compared to the US, UK goods were a bit pricey, a bit shoddy, and had a longer shipping route. And of course, Canada would be taking a big hit economically to join.
So. Unless the PoD is some sort of political Imperial Federation, as mentioned above, I don't really see this. An Economic unit (like the EEC) just doesn't really make much sense.