I selected the Greece scenario (not campaign) and played the Axis on General difficulty (the lowest setting where the AI isn't even more stupid than usual). All options selected: Weather (definitely!), Supply, Fog.
In short: I failed, getting a regular victory on turn 14. But read on...
The Italians live and die by their 105/28 (their artillery). The infantry is very frail, but as long as they're backed up by supportive fire they do alright. The goals of the Italians is basically only to survive. Without requiring neither the Wehrmacht or the Luftwaffe or the Regia Marina (the Italian Navy) to rescue them! But the trick is that merely by doing so, they will munch up the units in their area. As a secondary objective, they can take out the Yugoslav cities or perhaps conquer the western non-victory mountain villages.
Yes, that's right - the Wehrmacht will simply ignore Skopje, Monastir and Ioannina - these cities aren't victory hexes!
The Bulgarians first act much as part of the German army, until the Metaxas Line have fallen (the fortifications to the northeast of Salonika. Then the path to Athens gets so narrow there isn't room for the Infantry. But they do well on "active garrison duty" which means to stop any stragglers from reach flag hexes, and help the Italians cleaning house in the last part of the scenario. Their artillery, however, is much more valuable. After making a single shot to start off the scenario, I pull it back to Plovdiv, where I can replace the ancient Krupp piece with the 10.5 design (I assume) the Germans have so thoughtfully provided - as well as buying some wheels! (A truck)
This is because once the Alakmon Line (the fortifications shielding Kozani) have fallen speed will be of the essence - minimizing the number of turns needed to reach Athens.
The plan for the German army is simply to annahilate any enemy units on a straight line from Strumica to Lamia. (Obviously taking Salonika, Kozan and Trikkala on the way south)
The fallschirmjager will first group away from sight (they must be loaded one at a time; and Plovdiv Airfield is too far away to help). Perhaps on turn 4 or 5 they will reach Kalamata; a few turns later that area is cleaned out by the two paratrooper units with the help of massive bombardment from the Italian Navy. And then they move north just in time to help the spearhead take Athens on turn 12.
That's the theory anyway... Ahem.
This plan was shot to pieces straight away. The weather was horrible, with straight rain turns 2, 3 and 4. It actually rained so much I got two turns of muddy conditions (in Greece in April! I had to check; yes the probability of rain is set at 5%...)
The most profound effect wasn't that my air units had nothing to do, or even the reduced movement. I strongly suspect the reduced visibility that makes me invisible for the first crucial turns (when the AI still has prestige) affected its decision process. So - unlike what I have come to expect - it did something very smart, and pulled back its fighter unit, and filled Kalamata with defenders; artillery, several anti-aircraft as well as cavalry units.
The way the AI never met me in the air up north combined with the massive AA presence meant I had to postpone my airborne assault on Kalamata. Not only did I have to shoot the AA with naval fire, I had to wait until Salonika (and its airport) fell for my fighters to get the range they needed to provide cover.
That doomed the attempt. Turn after turn the paras had to wait, with no air cover, with more than one fully functioning enemy AA unit, and with a British Hurrican hiding away.
Meanwhile, in the north:
The Italian campaign went well (measured by Italian standards at least). Grouping around the arty, I easily defeated the assaults made by Greek mountaineers as well as a British Crusader. You lure them in by placing your units within their sight range, and they attack. Your arty cuts them in half and then you get one more shot followed by all three infantry going in for the kill. At least, as long as you can end up without exposing your precious arty, and without infantry ever leaving its side. Also, when the arty runs short on ammo, it's better to sacrifice a turn and resupply than risk getting your entire front unraveled. Yes, the Italians do need very patient babysitting!
I think the weather helped the Italians here - they weren't planning on going anywhere fast, and I think I got better odds on the Cruiser tank bogged down in the mud.
In the center-right: As expected, the AI moves Yugoslav infantry out of their cities to attack me in the rear. End result: one dead Yugoslav unit. I bypass completely the right-most Greek Metaxas Line defender - that's not the way to Athens. If it comes looking for trouble, my Bulgarians can handle it.
It's better to focus your fire, ensuring you kill off one defender good, than to spread your fire, risking leaving stragglers alive - stragglers that block your path!
The greek artillery is a nuisance, so keep it covered with a plane to reduce its ability to reinforce if you can't make a clean kill. Obviously you want to suppress it before directing infantry to attack the defenders its covering for.
Once Saloniki has fallen, with any luck the Italians can fall the Kozani defenders in the back at the same time as you ford Axios River. (It's a minor river, so you'll want to move units into the river not only at the bridge hex)
Then it's a race down south. Here your Sturmpanzer is invaluable, since it can move quite a bit and then still fire. Together with your trusty Stuka you must make sure the Larisa and Lamia defenders can't hold you up. And try not to detour your spearhead units - remember Trikkala does not need to fall before Athens does! (The road is so narrow so simple logistics will most certainly make you find that some units won't make it to Athens anyway. Use them for Trikkala!)
In my case the bad weather had turned the Kalamata landings into a nightmare. Just at the crucial time, the AI cleverly(?) committed its naval force. Once I had Saloniki it was a trivial matter to escort the Paratroopers, but bad luck meant that one of them landed on the clear terrain southwest of the city (the clear hex). Since they land already suppressed, and had nowhere to retreat, even a pesky Greek Cavalry could eliminate half my force in a single attack!
But the remaining Fallschirmjager persevered (not that it had anywhere else to go!), and all by itself cleaned out the cavalry, infantry and AA units. (Okay, so the naval bombardment helped a bit...)
But it was too little too late to help out at Athens. And the Kalamata disaster meant that the regia Marina had to stay at Kalamata too long, and couldn't bomb Athens for me.
End result: Athens fell on turn 14.
Now that you've read this, I expect you to do better than me...
) and your TAC Bombers to clear a path to victory for you.