A lot of time in alternate-history fiction, the fall of Winston Churchill and his replacement with Lord Halifax (or Halifax becoming Prime Minister instead of Churchill in the first place) is considered the death knell of the (Western) Allied cause. Hitler is left to dominate the Continent, with Britain eventually (or even immediately) doomed to become a Nazi satellite. This section of the Wikipedia article contains several such instances.
However, if you want a different look at what Halifax might've done as PM, there's a timeline on my alternate history forum simply entitled "Halifax." Halifax does seek to negotiate a peace with Hitler during the dark days before the evacuation at Dunkirk, but this is no supine surrender to the Forces of Darkness. Instead it's something like the Peace of Amiens, which ended the War of the Second Coalition against Revolutionary France. Everybody knows there's going to be a rematch--Hitler wants to secure his western flank before his long-awaited invasion of the Soviet Union and the Western Allies (Paris has not yet fallen and so France is still in the war) need time to reorganize and rebuild after the drubbing they get.
Some interesting bits of the timeline include the Nazi implementation of the Madagascar Plan, which is an improvement (but by all that much) over the Holocaust--Soviet Jews are still murdered en masse and Madagascar is basically one giant pit of deprivation and forced labor for German and Polish Jews. There's also the earlier deployment of British jets, which give the Luftwaffe quite a hiding, as well as a pro-Nazi insurgency (that in our history didn't really accomplish much).
If you liked "Halifax," the story continues with a sequel entitled "Clem's Vision." It seems to fizzle after three pages, but it does set the stage for some interesting future developments.