When Churchill became PM in 1940 the situation was quite grim for Chamberlain. He had always been unpopular with the british left. In addition the british had suffered a defeat in the Norway Campaign and had to abandon Norway to themself with some minor victories at sea. The Norway Campaign had lead to a debate in the House of Commons and in a subsequent vote of confidence Chamberlain suffered a moral defeat and the government only had a majority of 81. This lead to a genreal feeling in Westminster that Chamberlain had to go. The obvious choice was Lord Halifax. He was the former Viceroy of India, was an experienced politician and had a reputation for his sound judgement. There were a couple of problems though. He lacked military experience and was not very interested in war. He was also a peer wich meant he could not sit in the House of Commons. He also did not want the position at that time and told Chamberlain that he would be reluctant to take over the reins. The only other person in the party that could succed him was Churchill(the others were to young, to unpopular or to inexperienced). There were still a couple of problems with Churchill taking iver the reins. His involvment in the disasterous Dardanell Campaign and his support for Edward VIII during the abdication crisis was just some of his action who distanced him from the establishment. He was also considered a "man lacking sound judgement" and "not suited the highest office". There were still some reasons for why he was chosen. He was interested in war and fascinated and energized by it. He had both military and political experience and, maybe most importantly, wanted the office. At a meeting between Chamberlain, Halifax and Churchill Chamberlain first offered the office to Halifax, but Halifax felt that since he was a peer, Churchill would end up running the Commons and defence and Halifax would end up being a lame-duck PM while Churchill took effective control. He felt it better that Churchill was PM and while he acted as a restraining influence as a Foreign Secretary. Therefor they decided that Churchill was to become the new PM.

If you have any further questions i would be happy to try to answer them.

Holland, J(2015)The War in the West A New History Volume 1:Germany Ascendant 1939-1941London: Transworld Publishers