What if Lord Hailfax had become Prime Minister of England in 1940 instead of Churchill? How would this change British handling of the war? Would Halifax sue for peace after France falls?
First of all, welcome to the board.
The idea of Britain bowing out of World War II in 1940 has been done a few times on this site, but hear goes.
While it's true Halifax supported Chainberlain's appeasement polecy in the run-up to September 1939, Chainberlain (and his cabinet) were pretty badly discreddited by the time of Churchill's assention to the premiership in OTL.
Though there were voices calling for peace, I'd say that Britain fights on in some form, after all Churchill will still be a vocal voice in the cabinet (unless you have the infomous taxi accident he was in prove fatal).
That said, Halifax won't be the same sort of War-time leader Churchill was, even if he takes over in Churchill's absence, which in itself isn't a sure thing imho.
Someone did a really good Halifax in 1940 TL once:
https://www.alternatehistory.com/discussion/showthread.php?t=135299
Have a read-it's interesting!