Would Churchill still become prime minister?

Would Churchill still become prime minister if Norway successfully resisted Germany?

Probably yes; France still falls and Chamberlain remains a spent force, although Churchill's timing or immediate route to office may change.

France still falls in this counterfactual, so the political crisis that elevated Churchill does not disappear. The timing might shift, and success in Norway could itself strengthen Churchill's position. Chamberlain is also politically and personally spent after using appeasement to buy rearmament time. Churchill therefore still becomes prime minister, although the exact sequence may differ.