OTL the Statute of Westminster came in to effect automatically in Canada, South Africa, and the Irish Free State. Australia, New Zealand, and Newfoundland all had to pass legislation ratifying it (Newfoundland never did & ented up reverting to a crown colony). I don't think there's any way to delay Australia's ratification by more than a few years (especially given the Pacific War), New Zealand might wait until the 1950s. Facsism is not going to be very popular outside the UK, and a facsist government in London will probally spur the weaking of imperial ties just as surly as WWII did.
As for economic union; Canada will always favour ties with the US over rest of the Empire/Commonwealth. The same with defence. On the other hand keeping most of the other members in the Sterling Zone and some kind of common market is fairly plausible. India will still become indepentant (& likely be split into India & Pakistan) too, and I just don't see either country sticking with the monarchy very long so republican membership in the Commonwealth will still be allowed. Ireland will also become a republic, but I can easily see it staying in the Commonwealth. I don't know about Newfoundland; it's hard to see it getting dominion status back so it'll probally still end up part of Canada (or maybe a self-governing colony like Bermuda).
What you'll probally end up with is a British Commonwealth were most members (but not Canada) have close economic ties and currencies linked to the Pound Sterling, most members are monarchies in personal union with the UK, but some are republics (with the odd independent monarchy thrown in), and all members having their own militaries, some of which are in a Nato-type alliance with the UK. No common citzenship per se, but varying degrees of prefential treatment for migrants from one Commonwealth country to another, and long-term permanant residents from other Commonwealth countries being given political rights (like the vote) than aren't availible to unnaturalized foreigners.
All good points. I hadn't thought about how fascism in Britain would be received in the rest of the Empire.
I do like the idea of keeping the Sterling in use in most of the Empire.
Keeping Newfoundland in the Empire instead of Canada would be interesting. Especially after the British go fascist....on that note, in the OV, this occurred in the 1960s...Mosely took power in 1965. Think this should stay the same?