In 1940 the BEF is captured in France and in late June Britain as well as France signs an armistice with Germany.
Ten years later the Germans decide to finish the job and invade the UK.
1: What are the terms of the Anglo-German Armistice?
2: What does Nazi Germany do in Europe during the 1940's?
3: What does Great Britain do elsewhere during the 1940's?
4: How do the Germans divide up Europe?
5: How does the British Empire develop during the 1940's?
6: What do the Americans do?
7: What do the Japanese do?
8: What technologies are developed during the 40's?
9: How are the British Forces deployed to defend the UK?
10: What are the invasion plans of the Germans?
1: UK retains all of its territorial possessions. Germany probably pulls out of Norway (no need to hold onto it without a UK at war with them). France remains Vichified, some kind of protectorate established over the low countries.
2: Attacks the Soviet Union; Drives the Red Army to the Urals but remains stuck in a massive, Vietnam style quagmire of, well, Russian Proportions.
3: The peace deal is a very, very, bitter pill for the UK. While building up their army would be the smart thing to do, you've got Halifax as PM and he's probably neglecting the state of the Army and Navy in spite of the Crisis. One can only hope this is not the case.
4: They are still working on it. Scandinavia and Western Europe is kept generally the same to keep the UK happy. Italy is a big winner in the Balkans but mostly it looks like OTL 1943 down there. The Russian People and the Poles continue to fight a desperate war for their own survival. The Concept of Racial War means that there are probably more Poles in the USA then in Europe at this point. The Red Army continues to fight; Its Industry is squarely in Siberia at this point and they continue to produce war material. Soviet Resistance is extreme and motivated by fear of an enemy that can and will kill everyone out of convenience.
5: The British Empire begins to have a meltdown. Palestine and India both face large movements against the Empire, which result in internal police actions. Furthermore, many hawks in the UK either emigrate to the USA (such as the Half-American Winston Churchill) or face national disgrace for the "1940 Incidents". There is Considerable hope that Hitler's successor, (Hitler planned to Retire in 1950) will honor Hitler's agreements with the UK.
6: The USA eventually goes to war with Japan in 1942 over a surprise attack against Pearl Harbor. The USA had previously cut off Oil to Japan, and the Dutch East Indies followed suit. In a parallel decision to OTL, attack was the only defense in the mind of the Japanese. By 1946, the USA has fully mobilized its resources and crushed Japan. A single nuclear weapon dropped on a major Japanese City compelled the Emperor to surrender. The USA is essentially acting on its own against Japan, but it creates friendly states in Vietnam (They aren't giving it back to Vichy!!), Korea (Syngman Rhee) and Japan itself. The USA is engaged in a ground war against Mao in China, but Chiang Kai-Shek's fortunes are rising and people like Vinegar Joe Stilwell have made real progress in straightening out the country. Chiang might be highly corrupt, but the USA's decade long efforts and the commitment of its own troops means that Chiang looks to be solid. The Communist Insurgency is little more than a Fringe Movement. The USA, now building a nuclear arsenal, looks at Nazi Germany with considerable Distaste.
7: Japan fights and loses to the USA. They are forced to renounce war as a means of policy, but by 1950 have begun a considerable economic recovery. Japan expected merciless treatment from the United States, but this didn't happen and, much like OTL, Japan is protected by US troops. By 1950, Japan is likely to be facing its second set of post-war elections.
8: The US has the nuclear Bomb, as a result of a massive government project to learn about it. The US war against Japan probably leads to the supremacy of the Carrier in Modern Fleet Design. The USA has short Range Missiles. As close allies and friends of the United States, China, Vietnam and Korea enjoy many of these weapons technologies, although probably not nuclear technology. The USA also develops Television and the first computers. The Emergence of Muscle Cars and Fast Food Restaurants are also imminent. Finally, The Assault Rifle was developed in 1946 in time for operation Downfall, a technology that has been tried in the Chinese War against Communist Insurgents. US weapons technology has a considerable testing ground in China.
The Soviets have some of this as well, as the USA clearly prefers a Soviet Victory over a Nazi Victory. But the USA isn't terribly happy about the Soviet Union, even though Stalin himself was deposed and replaced with a troika designed to save the Russian People from the Nazi Butchers (The new leadership is committed to the survival of its people and are fully willing to co-operate with the USA's vision for a postwar world, in the hope that there will be a post-war world.)
Finally, Germany has wasted much of its efforts on absurd and poorly considered weapons designs. While the Panzer IX 'Krokodil' is a well designed replacement for the aging Panther III design, mass production of the Maus, and its slightly saner replacement model the Maus II, has turned out to be a disaster for the Germans. The 105 Ton Maus II, though slightly smaller and stronger than its predecessor, is still too large and too vulnerable to stand up to a prolonged battle. The third machine of the German tank forces, the KaiserTiger, is a maturation of the Bulky and difficult Tiger designs, finally given an engine strong enough to keep up in a mobile capacity and carrying an incredible 150' mm Gun.
Germany is the only nation in the world to have Intermediate Ranged Missiles, the V-5, but they suffer from low accuracy. Germany's aircraft are perhaps second only to the USA's designs, as they suffer from some degree of fantastical designs, rather than experimental testing. This is more than enough to hold off the Soviet Union's designs. Germany might have Nukes in a few years, but their entire nuclear program was botched or sabotaged. In any case, the German A-bomb effort failed to generate any results. A second attempt has probably been started by the New German Fuhrer, and its probably one that will see results in perhaps two year's time.
9: The UK's would assume that diplomacy will see its continued survival. Despite the increasing power of the Kriegsmarine, Hitler did continue his charade of only partial control of Western Europe. The hope is that these states will refuse a German buildup against the UK forces. The Halifax government has naively allowed the country to return to a peacetime footing. That said, the Royal Navy remains a powerful combatant, with much of the Royal Navy ready to avert an invasion however possible. The Halifax government has not been completely blind to an attack, however. The UK has a considerable standing army and a larger body of militia they would press into service if needed. Finally, the UK has a reserve of Chemical Weapons that would probably used if desperately needed. The Real hope, however, is that the USA will honor its guarantees to defend the United Kingdom in the face of attack. While the US army has been trimmed to defend Japan and fight communist insurgents in China and defend a few positions, a large of number of Pacific War Veterans are in its ranks and the USA has 30 nuclear weapons in its arsenal. In any case, the USA would probably need months to move its troops to the UK--and the UK's choice is that the Royal Navy and the Royal Air Force would survive long enough against the German Forces against them (Remember that at least half of the German Army is needed against the still-dangerous Red Army in the Urals) for the USA to intervene and save the day.
10 Germany's invasion plans also consider time. Unlike the UK, Germany is fully aware that it can mass its forces in Calais without triggering a war. The war would involve a Naval Blitz, against a Royal Navy of superior strength and a Royal Air Force of somewhat weaker power. While the UK gambles that such an invasion would take months and therefore the USA will be able to land throughout the country, the Germans plan on the opposite gamble--that the invasion will be over in two months. That advanced U-Boat designs will kill much of the Royal Navy, and a massive paradrop attack against the UK will will shock and break the British enough to allow the main force to steam inland. The Germans committing a large numbers of soldiers to the attack, and have plans to support them with air transport. The Germans fear the interference of the USA into the conflict, and would assume keep them out of Europe entirely. The Germans would also attack Northern Ireland and make a Deal with the Irish in exchange for their neutrality. In any case, the USA will have to islandhop all the way from Iceland to the UK--a very difficult attack that Germany hopes will lead to a Transatlantic Stalemate.