I don't see Churchill sitting by for three years without doing anything in Europe. While Britain may mostly be building up Bomber Command, he'll look for something for the army to do. I foresee several possibilities.
First, an invasion to liberate Norway. This definitely becomes possible after the German invasion of the Soviet Union.
Impossible. Against real resistance you can't take Norway through any port except Oslo, the other ports are all too small to support any major efforts. Even against a scattered politically divided Norway fighting with defenses dating back to the 19th century the Germans STILL got the shit kicked out of them and only subversion within Norway and their paratroopers made the invasion work. With all the forces Hitler poured into Noway its a complete nonstarter.
Second, placing an Expeditionary Force in the Soviet Union to help Stalin. While laughable 1943 and later, and unlikely in 1942, in 1941 it's possible Stalin agrees to a British force on a specific front. There are two options - one is a British force in southern Russia supplied through Iran to fight in the Caucasus/Ukraine. I think this is the least likely. The other option is to send troops through Murmansk to keep the polar Lend Lease route open and defend Leningrad. This could be combined with a Norway operation and keeps better control of British supply lines. Stalin may later regret allowing the British in, but he can always limit the scope of their participation once the crisis has passed.
This just needs one thing to make it happen: Stalin's Dead Body. Once his troops see what the British are enjoying as "field rations"...so much for "starving westerners" propaganda.
I think there is a real possibility for a "Northern Front" happening ITTL. It has all the Churchillian trademarks. An aggressive, dubious move to help an ally. Geostrategical possibilities to knock out a minor ally of the main foe (Finland - very real chance it'll drop out and join the Allies at some point) and possibly bring in a new ally (Sweden). It would repair the botch Norway operation of 1940, and provide new airbases in Scandinavia to strike at Germany's heartland.
This has all the trademarks of Churchill losing his job.
The LOCs up there make Finland untouchable and Sweden isn't interested in joining either side. This early in the war gaining air superiority is not only not assured, its unlikely. Where will Allied airpower be based? Not carriers. They'll all be in the Pacific or Indian Oceans. If the entire US carrier force in 1944 plus the USMC were made available, it STILL wouldn't be enough to defeat the numbers of troops, the Luftwaffe, the terrain, and even the Kriegsmarine operating in what would be very defensible waters.
Most likely scenario is London makes plans for an operation in Norway. They don't happen, but Churchill does send an expeditionary force to Murmansk in August 1941.
Does the term "PQ-17" mean anything to you?
Norwegian plans are revised accordingly awaiting for an opportunity. Britain has the main responsibility for defeating Finland so Stalin can move Red Army units away to meet the Germans elsewhere.
This is Arctic Swamp terrain that is NEVER, 12 months a year, good for warfare using the motorized infantry of the Allied armies. And there are NO appreciable rail lines running in Northern Finland. Just on the Baltic coast. There is a reason why the front lines up at Murmansk went little changed in WWII. Even the vaunted "all weather" capability of the vaunted Red Army were strained beyond the breaking point in that sector.
After Japan attacks, there may be a Torch analogue as US forces invade Norway in summer/autumn 1942 before the US Congressional elections.
Over George Marshall's Dead Body.
Depending on what kind of diplomacy happens and how bad the fighting in the mountains are, Allies have a Scandinavian campaign better, as bad, or worse than their experience in Italy IOTL.
Arctic swamp, heavy forest, mountain ranges, no rail lines, no major ports anywhere north of Oslo and Copenhagen in the Atlantic, much of this
north of the Arctic Circle, and virtually no campaigning season whatsoever (6 weeks, maybe?)
The worst day to campaign in Italy all year is still a better day to fight than the best day of the year in an arctic swamp. It would be not unlike trying to fight through the Qattara Depression in Egypt.
Now perhaps if Turkey can be convinced to allow British naval forces into the Black Sea, the amphib ops might be stepped up in 1943 with several British corps & a army HQ. Maybe commando ops if the ships can be provided in 1942?
After having their fingers receive 3rd degree burns in WWI no way was Turkey entering WWII. They only DoWed Germany on 2/23/45 because the Germans had been driven so far back that the nearest Axis controlled territory was in Northern Yugoslavia. They only did so to join the UN, and DoWing the Axis was a requirement for entry. If Russia had collapsed they would have probably DoWed the Allies just to save themselves.
No doubt Churchill would drawing big blue arrows on his maps of Scandinavia. Brooke, Eisenhower, & many others would have to hear out his ideas for striking at the heart of Germany via the 'Danish Bridge'
No doubt Ernest J. King would be drooling over the prospects of getting his precious "Japan First" strategy in the face of this Churchillian nonsense.
Sadly, its obvious Gallipoli hadn't taught Winston a damn thing.