The best course to prolong the existence of the Axis powers is to not invade the USSR until the British are finally and properly dealt with: Hitler's abrupt turn into Russia left a wounded but still fighting Lion at his back, and that would ultimately spell his downfall as, besides the British War effort as a whole and their early coordination with the USSR, the British were vital in supplying Allied agents on the Continent, serving as a launchpad for D-Day, and providing the crucial bomber bases that pounded the German production efforts into dust.
Assuming Dunkirk goes on as in reality (French defense of the Beachhead was one of the reasons the Germans failed to push through even after the halt order was lifted, so SOME men would still get away) the drastically weakened British military holes up and prepares for the "inevitable' invasion. Of course, Operation Sealion would never have succeeded, as the Germans simply lacked the maritime capacity for a seaborne invasion. The Battle of Britain instead goes ahead, but as the RAF wasn't so torn apart as the Army, they would likely still fend off the Germans unless the Luftwaffe made a more concentrated effort to disable the Chain Home Radar array, and even then, the shortcomings in Luftwaffe aircraft design(the BF-109's oft-cited short range, for example, and the limited bomb loads of the Heinkel and Dornier bombers) would still put the ball in the UK's court. So the Battle of Britain would likewise play out as in reality.
However, the Germans were still by and large "beating" the British for the first two years of the war, denying them any significant land victories and repeatedly driving them from the Continent (France, Norway, Greece.)
Refocusing their production on Aircraft and U-boats while offering the Italian fleet modern upgrades such as radar to close the Naval gap in the Mediterranean could have helped with the commerce war, as would ensuring the Kriegsmarine Surface fleet at the very least had a coordinated battle group instead of sending their capitol ships out in ones and twos. The Graf Zeppelin Carrier, while its prospective role is greatly overhyped, could have served as an excellent support vessel in the "Mid-Atlantic Gap" where Allied Air power was restricted in the early part of the war. Coordinated with the Bismarck, Tirpitz, and the Upgunned Scharnhorst and Gniesenau (fitted with 15-inchers, as was originally planned) could have struck the balance against British battleships, especially with aircraft from the GZ to hunt British vessels and protect against Torpedo planes
Of course, the Royal Navy would have eventually hunted down and destroyed the German fleet, but doing more damage earlier in the war could have tipped the Balance in Germany's favor, especially if they scared merchant crews into refusing to sail or forced the British to spend more resources protecting Convoys or hunting the KM, weakening their forces in the Med and Pacific.
Closer coordination with the Italians and Japanese and a concentrated effort in North Africa (sending more than, you know, three Panzer divisions) could have taken Cairo and the Suez Canal, cutting the British off from their Pacific holdings which the Japanese would then attack, forcing the Commonwealth to either withdraw their troops to defend their own shores (as Australia's PM threatened to do at one point, though his seriousness can be taken with a grain of salt) and likely leading to anti-colonial uprisings in India.
(Indeed, a particular German failure was their lack of effort to exploit the anti-colonial/anti-Soviet attitudes in the regions they occupied or wanted to deny to the Allies: the Middle East held a good deal of Pro-German sentiment, such as Reza Shah in Iran, and the Ukrainians and Baltic peoples welcomed the Germans as liberators from Stalinist oppression. )
With the British Empire torn to pieces and bleeding from every limb, the Parliament would likely oust Churchill and replace him with someone more willing to negotiate. Not Halifax, as he never wanted to be PM, and certainly not Oswald Mosely, as some people have suggested, but someone.