Well Chamberlain was dead in two years and currently 69, which is a lot 'older' in 1938 than now. So it's possible the cancer claims him early, or is felled by a stroke like his father. As for successors...
Hoare is 'the obvious candidate'. We have to remember appeasement was popular at this point, and he is an arch-appeaser. However, some folks won't like what he did with the French over Ethiopia, and he was noted to be a poor public 'performer'.
Simon is 'most unlikely'. He's a National Liberal so won't have any really power-base within the Conservatives [who will ultimately decide]. If he's elevated to PM, it will be from the desires of the Tory 'Grandees' [who are the electorate here]. They will not do so, as Simon is cold in manner, disliked and distrusted. At 65, is approaching the 'too old' stage. Plus, he is a Nonconformist.
Eden is 'the Young Turk'. Only 41, but charming, photogenic and hardworking. Officer in the Great War, popular in the country. Main minuses; is anti-appeasement, is somewhat erratic and is generally viewed as the 'idealist with no convictions'.
Hore-Belisha is 'impossible'. Charismatic, quick-witted, driven and a good innovative problem-solver. At 45, is also part of the 'new generation'. But carries three crippling defects. One, is a National Liberal. Two, is anti-appeasement [though less than Eden]. Three, and a big three - is a Jew. Anti-semitism had also attached the air of him not 'really being one of the chaps'.
Cooper is 'the dark horse'. Rather like Eden, but has more substance behind him and is more 'one of the chaps'. Noted natural leader, is a war hero etc. Unfortunately, is also seen as dissolute, indiscreet and a bit of a firebrand. Despite these minuses, the grandees may select him if Munich goes all sour and they'd like someone kinda like Churchill, but not Churchill.
Inskip is the 'emergency placeholder'. Staunchly religious, lawyer of long experience. Acceptable to the party in general terms, doesn't offend anybone due to being a well-known nonentity.