Funny, because BBC America really did have content that was exclusively British. Then they found they can make more money airing Star Trek franchise and other USA content. It seems Graham Norton is the only truly, fully British show offered on BBC America (and turned around in just a DAY, I believe), and Top Gear had its implosion and subsequent implosion, so it too MIA.
You have a great point, but I fear the only place you can find of what you're looking for is with OTT like Hulu or Netflix. They do have some Aussie and Canadian content or English language international content, of course, it is entertainment, not news and public affairs.
What is somewhat sad is that even the English language versions of foreign channels are really more about promoting or highlighting that country's culture--hardly any entertainment programming--and the news is very USA centric or is caught up in trying to provide that country's view in international affairs rather than a great newscast or show. The foreign language versions can have really great straightforward newscasts and news shows like TV5's news and information watching, while France24--an English language service--is just a mess.
Some Public TV stations (NOT offered as part of the PBS National Service offerings) are airing more and more programming--entertainment--from Australia and Canada, but those are offered from 3rd parties, so it depends upon your local Public TV station willing to pay and air that content. Otherwise, OTT is your best bet.