This is really funny, seeing our recent lives portrayed as history, only possible to understand through archive material!
Cool Britannia was part organic (Oasis, Blur, Pulp - all bands with very distinct English place-based identities), part manufactured bandwagon-cashing-in-upon tosh (Spice Girls, the better quality All Saints). It was a combination of luck, that a number of good bands came through at the same time and a reaction against the American dominance of early 90s Indie, Grunge and other cool genres. So a moment of Britain reasserting itself culturally and economically - with the Blair Govt sweeping in 1997 - after a tough few years and against the American cultural and economic behemoth.
Nothing to do with the Commonwealth. Everything to do with plucky little Britain displaying cultural capital that far outweighed our declining political and economic influence in the world, in the context of America, China, the emerging BRICS (had they been identified as such then?) and incidentally showing off how much cooler we were than the rest of Europe- but in pop music terms that’s never needed saying.
It all contributed to making London a very cool place to visit and live. It wasn’t just music either, there was heightened emphasis on culture, including the YBAs (Young British Artists) for example - look up the Sensation exhibition and hype around that.
What did we think about Australia at the time? Big, hot, full of beaches, sport, BBQs and kangaroos, bit of a cultural desert, good place for a gap year if you could tolerate the sexism and racism.
Culturally, more cheerily, Australia had given us Neighbours, Kylie and Jason in the 80s (not cool but very cheerful, Kylie didn’t become cool until later) and thousands of short-visa bar staff in the 90s. So the Australians we saw were young and fun. They contributed to the London buzz.
Did we think much about post-colonial government and nationhood across the Commonwealth? No.
That was and is an internal discussion for Australia, Canada, Jamaica, NZ etc. I think the only change is that now, when I explain to a child that the head of state of Canada or New Zealand is King Charles, it does seem surprising, to me as well as them. In the 90s it seemed anachronistic, likely to change in the next decade or two (none of our business) but there will have been a closer living legacy of connection and interdependence, from WWII and after. Less so now.