>>213585659alt-history is always difficult if you have an autistic interest in the events, because the point between reality and fiction gets blurred, and it's increasingly difficult to calculate potential outcomes based on realism.
Obvious disclaimer that Hitler would never have nuked Britain as he kept hoping they would see reason and ally with him, and also alienated all the best nuclear physicists.
Somehow a breakthrough happens, and Hitler has a nuke. You have to consider what the production capabilities are, is it just one nuke, or will there be more?
Assuming it's just one nuke, it would be used on his arch nemesis the USSR, in a hope to really "kick in the door" and bring the whole structure down.
He might use that nuclear strike to try to make England back down.
If If there was a production line, they would also probably test it in the east first, and then threaten Britain.
Trying to envision a scenario were Britain is threatened by nukes and doesn't sue for peace is difficult.
The Japanese were the most zealous and fanatical of the Axis powers, and they surrendered immediately upon being nuked.
Let's say it's '43 after the invasion of Scicily, maybe at that point Hitler would be forced to utter an ultimatum: Back off or I start removing your major cities from the map.
Churchill calls his bluff, or doesn't agree with the power of the Nuke, or hopes the RAF will keep German bombers at bay.
They get in, drop the payload, and remove London.
The ramifications are impossible to calculate, as you have so many permutations to go though. Did they remove the Allied high command? Did they kill some of them? Did some escape? Was the city evacuated, or was that never even attempted?
Sticking to logic at this point becomes basically impossible.
Thinking about the event, and the immediate aftermath by themselves, I'd think that Britain collapses, unless Churchill wasn't hit and is able to consolidate power in Scotland or something.