In 1940, one of the great jokes of WW2 is that Churchill becomes PM because Chamberlain stands down after surviving a confidence vote over the Norwegian Campaign with a smaller than expected margin and large scale Tory defections. The conventional histories see the Norway debacle as being Churchill's fault as First Lord of the Admiralty. So his own botch makes him PM.
The Norwegian MuddleChurchill as warlord was at best an acquired taste. He kept strange hours by military standards [up all night and slept half the day]. He came up with new schemes a mile a minute and had no respect for either traditions or bureaucratic work flow. His military commanders found him impossible to work with and a source of chaos. However in reverse he found the aged commanders and endless committee muddle exasperating beyond measure. In the particular case of Norway he was more right than wrong.
The entire Allied response was perpetually two days later and two pounds short. Now let us presume that Chamberlain gives him more of his head on the Norway planning just to be rid of him from the main deliberations of war cabinet. I will only make a few changes. First he's allowed to get commanders able to work with him [Cunningham for the Navy, Alexander for the Army]. Second the fleet puts to sea two days earlier and with a clear plan to force landings in southern Norway. In OTL the British paralleled the German move to land all over Norway when the key was breaking the air link from Denmark to Norway.
A new article by Scott PalterSo Cunningham destroys the bulk of the German Navy [BC Scharnhorst, Gneisnau; CA Blucher, Lutzow; CL Koln, Konigsburg, Karlsruhe, Emden; 24 destroyers, light ships, transports and tankers]. He loses three combatants [BB Renown, CV Glorous, DD Glowworm] but the bulk of the carrier air group gets ashore to secondary Norwegian bases. The four British brigades with Alexander retake Bergen, Kristiansad and Oslo with Norwegian help, isolating the Germans at Trondheim and Narvik [as well as the heavy cruiser and 14 destroyers who landed them]. Alexander was no field genius but he was adequate [see Burma in OTL]. The fleet also lands an RAF fighter wing which uses the retaken Norwegian air base at Fornebu.
This creates a stalemate. The Germans can throw in more planes but they are operating at range while the British and the few Norwegians are not. Also pilots lost by the Allies can be saved while the German aircrew is lost. So the Germans have the advantage in an attritional contest but never quite gain air superiority. They sink a few lesser British craft and provide some support for their scattered and fleeing ground troops who are over the next two weeks mostly captured or chased into Sweden where they are interned. In the meantime Alexander gets 2 more British brigades, 2 small French ones and a Polish brigade. With these plus the rapidly mobilizing Norwegians [they have little in arms and less training but are quite willing to fight for their homeland and are more adept dealing with terrain and climate than their allies] he is able to push north while holding the south. Cunningham has in the meantime stayed at the fringes of German fighter range and mopped up the remaining German naval forces.
By the third week of April the Germans have to start shifting their air units for the main offensive. So Alexander and Cunningham are left free to slowly push north to Narvik chasing the two German garrisons into Sweden.
So what does this change? France still falls. The air and ground losses for Hitler are small change other than the paras. Those lost in Norway are unavailable to be pissed away at Rotterdam. So Holland takes two more days to fall. This is actually bad for the allies as it pins the very good French 7th Army at Breda for that period of time. In theory the French 7th was the motorized reserve that could have stopped Guderian. However in fact the French wasted this chance in OTL so it's a wash.
The big changes start when France falls. Sea Lion goes from a long shot to simply impossible and both sides know it. No German surface fleet left and no paras. So the ONLY way for Hitler to get at Churchill when he won't make peace is to go south. Instead of the Battle of Britain we have the Battles for Malta and Egypt. British will lose Malta in August and Egypt in November. A full German air fleet in the Med and a large panzer corps [all the Libyan ports could support in addition to the Italian 10th army and air units] is enough to bash the Western desert Group out of Egypt. British drop block ships in the Canal. One division [make it 4th Indian] retreats up the Nile for upper Egypt and Khartoum. The rest retreat back into Sinai after totally wrecking Alexandria and Cairo. Canal is now littered with block ships.
This sort of campaign does not produce a vast German campaign beyond into Iraq. Without Alexandria as a functioning port the limits of how many men can be supported in Egypt by truck from Tripoli and Tobruk is quite limited. In OTL it took more trucks to support Rommel than to keep an army group running in Russia. So Iraq would be at the expense of Barbarossa which I do not see happening. The probability is spending what supply can be brought forward on rebuilding Alexandria [rebuilds were never a Nazi strong point - no glamour], entrenching along the Canal and a push up the Nile to try and save the Italian army group in East Africa [hopeless but an Italian priority].
So the imperial disaster that will trigger the Quit India movement becomes Egypt not Singapore. In reverse the fall of Egypt means that Malaya and Burma are adequately garrisoned by late 1941 making the initial Japanese blitz near impossible. Similarly the Greek campaign does not happen because Benny the Moose has too much to do in Egypt and Sudan. A whole different war because the British fleet sails early and is better commanded.