So from what I can tell, a lot of issues come from a lack of modularity/uniformity between guns, tanks, trucks and more when it comes to the German military. Standardising German production would help significantly reduce the strain on the German industry in the long term, making newer and older models able to both use the same type of ammunition, tank parts interchangeable, etc.
The Italian military and Axis at large also suffered this as well, here's a good video on what I'm talking about. If the Germans made up their mind on allying with Italy sooner or at least investing in Italy. German investors and German trade agreements, advisors, joint-training, etc could solve or at least alleviate many of Italy's own issues.
German investors/industrialists/advisors could help develop the Italian industry much faster and better, German technology (for guns, artillery, tanks) would easily outperform the Italian guns we saw in the war which were commonly discarded in favour for older WW1 rifles. Doing this for minor Axis powers like Hungary and Romania (even if later than Italy) would help a lot of the issues. Germany throwing his weight and soft power around to standardise the Axis' equipment.
Standardisation of equipment and perhaps even joint-research would significantly help with the equipment situation, make the subpar Italian troops much better due to their poor equipment being answered. Likely without the equipment shortages immediately visible on the war's start due to the newborn Italian industry, now fixed or alleviated by German investors, industrialists and German-Italian cooperation.
Germany subtly assisting, or perhaps even outright assisting the Finns in the Winter War rather than the Molotov-Ribbentrop pakt and choosing to go through Polska and continue into the USSR after a brief pause in Polska to prepare trains to transport gear to the frontline. Which, while implausible, could be interesting since the USSR would've just started its invasion of Finland and shown itself a fool. Jet fighters more used by 1942 could be interesting.
Even if it's just support and an exchange of commanders, strategists, etc, it could prove invaluable for the German war machine to begin stockpiling equipment and training with Finnish tactics in mind as the Winter War progresses in preparation to betray the Molotov-Ribbentrop Pakt. While not war-winning, in tandem with alleviating the equipment situation-could prove useful as snow becomes useful to the German soldier with Finnish:tm: skiis.
Tighter control of the SS and information by Herr Hitler and a diversion of SS funds to infrastructure and supply trains rather than death trains could free up a lot of resources, lower the amount of Partisan resistance. Save the liquidation for a later time, just (by NatSoc standards) starve them out at worst-the answer to the "Jewish Question" can come after the 'glorious Reich is victorious' after all. If not an outright delay, perhaps a gutting of funding to make its effect less known, since the German high command's fumbling as to what they should do could probably be exaggerated to delaying the question until a later date. Perhaps without liquidation camps and trucks, Poles, Ukrainians, Russians, etc would be more likely to join and fight for the Axis.
Like donanton mentioned, don't trust neutral nations like Sweden or Switzerland with anything-intel or otherwise. Perhaps it'd prevent a handful of disasters that could prolong the war and make the Reich more effective and speaking of foreign designers-perhaps the Germans could take more cues in designs for tanks from Czech and French counterparts.
Back to allies again, Italy. Italy outnumbered British forces in Egypt 10:1 - their binary divisions, while more mobile than the larger infantry divisions, were still easily outrun and surrounded by motorised British forces. A better Italian industry could mean better armed, more conventional divisions and perhaps even motorised battalions to assist a push into Egypt. If all goes well and Egypt is occupied, a push into Sinai and the Middle East could prove devastating to the British and very useful to the German war machine. Iraq seemed to teeter into pro-Axis alignment and even if neutral, could have her oil transported into North Africa and to the Italian fleet, army and also the Germans.
Speaking of oil, concentration on earlier synthetic oil factory, further development of Romanian oil fields and a secure line of trade with partners like Iran and Iraq could significantly help to oil the German war machine. Even considering all these factors and assuming the Axis is competent enough to make use of these, D-Day would likely still happen (Even if repelled, it's essentially an inevitable defeat with the USSR forever in the East and the Anglos in the West). So while not war-winning, could reduce the casualties significantly for the Axis and keep the USSR mostly out of Eastern Europe.