Doesn't have to be one of the War Cabinet. Britain is a democracy. Conservatives are the largest party in the National Government. Anyone who can command the support of the Conservative Party and, ideally, maintain Labour and Liberal support also. Labour didn't like Hoare mainly because of the Hoare Laval pact and Leo Amery was compromised because of his son's collaboration with the Nazis. Don't think anyone seriously considered Kingsley Wood as a war leader. David Margesson was a tough minded individual and Chief Whip. Labour wouldn't love him but no public reason to refuse to work with him. Euan Wallace did a good job getting the British forces to France and around Dunkirk too (he would die of cancer in 1941 but no-one realises that in 1940 including himself. Lord Swinton another very capable man. William Ormsby-Gore (Lord Harlech) an anti-appeaser of proven credentials who resigned in 1938 and a man of some ingenuity and intellectual capacity (you can still obtain his detective novels on Kindle -he wrote as William Gore).
And then there is the option of a politically neutral "expert" serving as Prime Minister -Lord Woolton, Sir John Anderson, Lord Trenchard, Lord Hankey.