I was only addressing Scandinavia or Norway.
As for trampling in France. it really depends on who is governing France. Petains attitude was France first. His policy might be summed up in Darlans response to Ambassador Leheays question: 'If you come with three divisions we will fight you, if you come with twenty we will join you.' Theres various ways to interpret that, but one way would be; 'If you are serious about defeating Germany we're in.'
The Brits had a small training base near Murmansk established in late 1941. If Stalin allows then building on that they could have in the spring of 1942 fighter cover to the eastern end of Norway. Hypothetically that could allow leapfrogging west & south, creating two or more enclaves on the coast and starting the isolation of German garrisons. Its a complex military & technical problem. If Dill remained CIGS I could see this at least getting a serious look. Its also the sort of thing that appealed to Churchill.