Regardless if the shooting war between Britain and Germany comes to an end because Churchill is dead, there is not a great deal of difference in the outcome. Despite the false premise on which this scenario is based, ie that Churchill was responsible for the continuation of the war after the fall of France, certain things dont change. Britains policy for at least 400 years, regardless of the quality of its leadership, has remained remarkably constant in that time....no single nation on the continent be allowed to dominate the continent. For that reason, Britain has always backed to the second most powerful nation on the continent.
In the scenario of a Britain at peace with Germany, (and presumably the other German lackeys like Italy), Britain will simply be free of Uboats and other constraints. presumably Japan will not go to war in this scenario, so massive aid via Vladivostock, and from the south is possible, in which Germany can do nothing because she is at peace with Britain. From July the Americans can join this push. Lend Lease to USSR (presumably both British and US) will be 10-20 times what it was historically in 1941. The latest and best a/c, unlimited amounts of food, transport, raw materials.....it will be all over by the end of 1942 in that scenario. Germany cannot raise additional forces, because the implicit threat from a continuing existence of the British, now free of a Uboat menace will require additional coast garrisons throughout Europe.
A return to normal trade conditions for Germany would never be accepted, except if the prewar demands like return of Polish security and a relinquishing of the occupation of western europe was to occur. Even with a truce of sorts in place, it was never even considered by the most rabid of the Germanophile members of the Govt to relax the Control commission Activities and a return to free trade for Germany. That had not been fully available to Germany since Munich. If it were Britain would effectively have surrendered, and that was never going to happen, even under Halifax. People should take the time to actually read what Halifax was prepared to do, before making such wild assumptions in their scenario constructs.
Halifax was a member of the appeasement group, and recently declassified documents of the British govt, show that in 1939, shortly after the outbreak of war Halifax helped with the travel arrangements of John Lonsdale Bryans, who believed he could bring down Hitler by making contact with prominent anti-Nazi Germans including Ulrich von Hassell, the former German ambassador in Rome. Initially Lonsdale Bryans thought he could drum up support for an anti-Nazi coup in Germany. But he subsequently changed tactic and tried to contact Adolf Hitler in a bid to negotiate a peace. These discussion went nowhere of course,m but Halifaxs association with the appeasement lobby meant that even if Churchill had not been on the scene, Halifax was most unlikely to garner the support of parliament to form govt after Chamberlain, another member of the appeasement group, so patently lost support. Had Halifax machinations with Bryans, which were known and fully documneted in the Secret Service file being kept on him and Halifax was to be released. This would have destroyed Halifax if he had any notions of making a grab for power. He knew the administration knew about his little jaunts with the enemy,m and that they were prepared to use them against him if he tried anything. For that reason, the so called halifax option was a complete non-starter.
In the lead up to the fall of the Chamberlain administration, it was becoming apparent that within the government itself there was failing support for the appeaement group. 81 abstentions, and on the 8th May despite a valiant attempt by Churchill to save his boss it was clear, his days were numbered. In desperation Chamberlain called a meeting with Churchill, Halifax, Chamberlain and the opposition leaders Attlee and Greenwood in attendance. The opposition was asked if it would support some form of unity govt. It becomes a little murky as to details after this, but the general consensus is that they came back with the reply that they would support a unity govt for the duration, but the tories had to rid themselves of the appeasement dominance. That meant that both Halifax and Chamberlain, along with other lesser fish, had to go. Halifax had been offered the prime ministership, but turned it down, knowing that he too could not gurantee the support of his own part or the opposition. British resistance was working independant of Churchill, though Churchill became the perfect man for the job of channlling and focussing that oppisitiion in the months and years that followed. There was never the slightest possibility of Halifax getting the job and Churchill was irrelevant to that outcome. If it hadnt been Churchill, it would have been another anti-German who took on the job, never was it a possibility of Halifax, or any appeasement oriented politician getting the leadership job after Poland. Ironically, it might have been possible, only if the allies were winning from Poland on. in a way, hitlers runaway success ensured the British would never surrender, whilst Hitler was on the scene.
There will be little difference to the outcome, except that the whole of Europe will be under Soviet control by 1945, unless the western allies respond, which they would do. They would never have allowed either germany or the USSR a free dominating hand in western Europe