Atvar was almost definitely bluffing, considering how reluctantly nukes had been used. Not to mention that the Race did not possess a limitless supply of nuclear weapons, and almost certainly not enough to destroy an entire continent.
Well the Race wouldn't have to blast the whole continent as they already controlled Poland, Italy, Spain, and a good chunk of France and presumably they wouldn't be nuking British or Soviet territory if their goal was just to smash the Nazis flat.
I would also argue, however, that Germany is as vulnerable as Britian and Japan, as most of 'its' territory is composed of puppet states that would quickly revolt if the German heartland were destroyed (which is one reason why I find it difficult to understand how Germany retained its sphere of influence after the Race-German War - Britain and Finland slipped the net, to be sure, but perhaps the threat of nuclear retaliation a la Bucharest might have kept the rest in line).
I agree with you that realistically the Nazis are just as vulnerable as Britain and Japan. The difference is Hitler is deranged enough to think that Germany can win a sustained city for city nuclear exchange with the Race. Churchill isn't nearly that crazy (and even if he was Parliament would never stand for sacrificing London just to try and get Australia or Singapore back), so he has much less leverage for engaging in nuclear brinksmanship.
However, I agree that Britain and Japan were much more vulnerable to nuclear attack than the other human powers. But I cannot believe that a British nuclear arsenal (and thus full diplomatic recognition) would not result in the retention of some overseas territory - South Africa, perhaps, or a part of India. Still, the UK got a pretty good deal at Cairo, all things considered.
India is pretty much impossible for Britain to retain by this point even if the Race was willing to give it up. After all Britain couldn't hold on to India IOTL, and the British are in much worse shape in WorldWar. (The British Isles were directly invaded, so it probably took a lot more damage than it did IOTL, it hasn't gotten any meaningful Lend Lease since 1942, and it won't be getting any Marshall Plan aid since the U.S. is going to be needing all its resources to rebuild itself.)
South Africa could be possible to retain but again would Churchill be willing to risk London, Birmingham and Manchester for the chance of holding onto South Africa?
With regards to a Japanese nuke, I can't see them passing up the opportunity to get Korea back.
Well in the novel line, Japan wasn't able to build nuclear weapons until 1965. Even if the Race lacked the ability or willingness to nuke Japan flat in 1944, it certainly had both the ability and willingness to do so by 1965, so Japan could no more extort Korea off the Race in 1965 than the Nazis were able to export Poland from them that year.