I am sorry, I don't mean to be rude, but that situation doesn't seem plausible to me at all. Italy in a much stronger position seems very interesting even if a bit extravagant.
But then starting from Greece and Yugoslavia leaning towards the "Axi"s is absolutely out of the question. They both have nothing to gain and everything to loose that way. I know that Yugoslavia really joined it, but we all know that the coup just days later brought it back to the "Allies", so this just proves the rule.
Then going back to Italy itself - it Rome we got here is so stronger then that also prevents Hungary, Romania and Bulgaria altogether from joining Berlin. The reason why they joined Berlin in real history is exactly the italian weakness. As we know fascism at the time wasn't considered as bad as it is today. In fact it had many sympathisers. We need to remember that Italy was doing better economically than the traditional democracies at the time, there was no antisemitism (i.e. Mussolini's Jewish girlfriend) and no concentration camps. The moderate people in Europe were actually more mad at France and Great Britain for not constraining fascist Italy and nazi Germany than mad at Rome for intervening in Spain. Hungary had excellent connections to Italy and so had Romania (i.e. common in a way roman culture) and Bulgaria (to mention that the Queen was Italian speaks for itself). Stronger Italy would instantaneously bring them very close to it and tensions over the Alps would result in neutrality or at best moderate friendly opinion towards the Reich in those countries.
If Germany has attacked Poland in that timeline the way it has in our world then the situation would be pretty much like in our world with Phony War in the West, USSR taking half of Poland, BUT Germany being on its own would make it think twice about Barbarossa, especially after a border crossfire with its southern neighbour and Rome going strong. Paris and London, together with Amsterdam and Brussels would be even more static in military actions, while of course militarising faster than what we have seen back then. Which is actually what I see the Reich like (after taking Czechoslovakia and half of Poland, doesn't really matter that much with or without Denmark and Norway, but in case Berlin leaves them be then that would even make the Phony War more, lets say phony). The Soviet's position would mirror that of France and Britain. So we have a balancing triangle in the West and another one in the East.
That would be like a carte blanche for Italy to go for Yugoslavia and Greece, partially due to the Allies giving a try to appease Rome by granting the Western and Southern Balkan Peninsula to it and partially to the belief that otherwise Rome and Berlin would go to war over them effectively knocking each other out (prety much like our history's nazi-soviet antagonism) or even better for the Allies - free for all between Rome, Berlin and Moscow.
So then Italy would go for Yugoslavia first. It doesn't matter if it wants to keep the whole state for itself or it is in alliance with Budapest and Sofia or not. The moment it crosses the border Hungary and Sofia would jump right away at Yugoslavia. From what I know Hungary was modernising its military at the time with italian technology. Bulgaria was also doing that showing great interest in equal parts in german, italian, czechoslovak and polish military hardware. The bulgarian military was pressured under the Peace Treaty of Neully (1919) to 30 000 men including the gendarmery, but that was rendered irrelevant right from the beginning of the 1920s by forming the Labour Troops (declared to be activated in only infrastructure projects, but actually that was a covert military) and the actual military of 30 000 which had to be professional was that only on paper. Every professional soldier in the documents was actually a training capacity and a private undertook military training for 3 or 4 months, then came the next without that even being recorded. So around 1940-1941 Sofia could field in a short notice an army of some 350 000 - 400 000 men. The only reason why the country fought the Balkan Wars and WWI was for ethnic bulgarian territories, occupied by the Ottomans and later by Serbia, Greece and Romania. So do you really think that with France and GB being neutral towards Italy invading the very same countries (which occupied territories claimed to be ethnic bulgarian only years before and which ITTL are leaning towards Berlin) Sofia would miss the chance to take them back and being part of the "good guys team" for a change in the process? Not a chance. The whole attitude amongst bulgarian men would be: "Our great-grandfathers failed to free our brothers and sisters (the war of 1877-1878 and later the Berlin Congress), our grandfathers failed (despite winning the Serbian-Bulgarian War of 1885), our fathers failed (despite winning their battles and being stabbed in the back from all directions). Enough is enough, here we are, willing and able, we want revenge!" Just to give you an example there were scenes in Bulgaria both at the beginning of the Balkan Wars and WWI when the telegramme for mobilisation comes at say 1:00 at night but the mayor thinks that he better gather the men first thing in the morning say at 6:00. But the population finds out about the telegramme, goes to the mayor's home and beating him up because he didn't declare it right after receiving it.
So Italy invades Yugoslavia, Hungary and Bulgaria do it right away with or without Rome's consent. In that case Mussolini would claim they are allied to him so he doesn't loose face for failing to keep second-rate countries at bay. So the whole of Yugoslavia is off the map in no time. Romania grows extremely nervious as it sees Budapest and Sofia both on the offensive and enlarged, but doesn't cross the border, because, you know, what about the USSR. Besides In that scenario it would still have Southern Dobrudja and most of Transylvania, and probably still losing Moldavia and Bucovina to the Soviets, so better keep what it has for the moment. We would also have Greece (axis-allied in the beginning was it, right?) fully panicked and feverishly trying to get closer to Paris and London (as is the actual tradition we know). After territorial gains in serbian and croat lands Budapest would be pressured by Berlin to distance itself from Rome, which makes an eventual magyar-romanian war even less likely. A growing tension in Transylvania nevertheless would once again constrain Romania from invading Bulgaria (not that the Romans would want to) as Hungary and Bulgaria have been allied to that moment. So now we have Italy going on Greece and still having Bulgaria as an ally, likes it or not. Turkey, having lost WWI, its empire, the aegean islands and having experienced a greek invasion would stay neutral while lurking at greek territory. So once again we have a state knocked out in no time. Bulgaria gets its territories back (Macedonia and its southern coastline), Turkey takes the easternmost greek territory and half the aegean islands. Italy keeps the rest a client state (minus Corfu, Kefalonia, Crete etc.) capitalising on the common heritage of both nations etc. etc.
At that moment we also have Bulgaria pacified. Italy would tell Paris and London that all it did was to prevent another major war starting from the Balkans. France and Great Britain would be in a much better defense shape at that time than in real history, but still no one knows what happens next:
- German aggression in Scandinavia
- German-Italian War
- German-Soviet War
- Soviet scramble for the Balkans
- Italian scramble for Africa
- intensified Italian industrialisation
- something else.
So the way I see what would follow would be either German aggression in Scandinavia; German-Italian War; German-Soviet War or all three in short time maybe preceded by a "cold war" of 2-5 years.
Unknown factors would be Japan; USA and the colonies of the British Empire. On the positive side the Italian industrialisation would bring much better hardware and the sought for carriers mentioned in the scenario (but only if Italy goes to building itself an African empire). USA would be more neutral towards Europe, especially when we see four different camps on the scene. The Americans would be like: "You see, there is a war of each against everyone, Monroe was absolutely right. They went through WWI which we brought them out of and they still don't get it." At the same time the situation in Europe would neccessitate that Britain keeps its ressources for home island defence. So we see rebellious Iraq, with probably Turkish and Soviet involvement instead of the German we know, rebellious South Africa, rebellious India, which the Indian Army would subdue, but that would also mean rebellious Malaya aswell, gaining independence and more aggressive Japan.
Result would be less devastating European front and much more vicious Pacific-Asian one, earlier end of the British empire, the Dutch East Indies, Soviet involvement in the Middle East and Eastern Europe, which leaves it ready for a German invasion, but stronger Japan in China or Soviet involvement in China, backing Chiang-Kai-Sheek while still supporting spread of communism south of China. Both ways USA would concentrate on even bigger navy and the final result would be devastated Japan, but without Pearl Harbour, Hiroshima and Nagasaki, and probably communist Japan in the end.