I don't think that a "full surrender" ala France was even possible. Suppose the UK is completely under Axis control and Churchill orders complete unconditional surrender. Do the Canadians simply mothball the soon-to-be 3rd largest navy in the world, stop producing war materiel* and paint themselves Axis grey on the map when there isn't a wermacht soldier within a thousand miles of our borders?
No, we align ourselves with big brother to the south and, taking a page from Mother England, we keep calm and carry on. The same would hold true for Australia/New Zealand forces.
* The UK had entered the war with 80,000 military vehicles of all types; however, 75,000 of these British vehicles were left behind in the evacuation at Dunkirk in 1940. We made good the losses - Canadian industry produced more than 800,000 military transport vehicles, 50,000 tanks, 40,000 field, naval, and anti-aircraft guns, and 1,700,000 small arms.
The UK might have surrendered, but the empire would fight on.
This is what makes the most sense to me. If the British isles would have been occupied, whoever authorities still in charge there would of course have "officially" surrendered - but Canada, Australia, New Zealand and India would not have followed suit.
The only thing we could reasonably debate is whether any UK "government in exile" would still have been acknowledged by the rest of the Commonwealth countries as actually being in charge of any coordinated efforts.