If there is peace between the British and Germans, then there is a general peace between Germany, Italy, France, Britain, Norway, Netherlands, Belgium, etc. The entire war is different. In fact, the war is over. Until Germany invades the USSR, there is no WWII and historians might consider that an entirely separate war.
In such situation, Italy probably gains something in the peace negotiations. If Italy is not given a portion of France's colonial empire (or even if Italy does get a bit), then there is probably some agreement among the great powers that Italy can do whatever it likes in the Balkans or at least agreement on certain demands the Italians will make.
In other words, I doubt very much Britain will involve itself in any war against Italy. If Britain does so, it upends the very peace settlement it just signed.
Of course, there may not even be a war. Mussolini decided on a whim to attack Greece because he was upset that Italy had not gained anything from the armistice with France, and that Germany told Italy to not do anything in the Balkans, but then sent troops into Romania anyway. He invaded Greece to give Italy a quick victory and hand them something.
In this scenario, it is unlikely the same situation exists. Italy will have gained something in the greater peace so his pride is assuaged. And depending on what Germany is doing to prepare Barbarossa, Hitler may even give Mussolini some indication what he plans and therefore Mussolini won't get upset over some Balkan intrigues. In any case, he has very little reason to invade some country out of spite.
In terms of Germany invading the Soviet Union. It is hard to say. Stalin really gave Hitler the benefit of the doubt. If Hitler comes up with some kind of plausible explanation why he isn't demobilizing, Stalin could still fool himself.
Even if he doesn't and knows an invasion will happen, Germany will stay do well initially. Germans may not get complete surprise and go up against some initial solid defense lines, the Red Army is still recovering from the purges. Stalin will still make the same kind of stupid mistakes he did initially. The Germans should still make some solid progress before winter sets in.
Of course, it's obvious that the USSR won't collapse just like IOTL, but this time the Soviet Union is alone. There are no other fronts for Germany to contend with. There is no British blockade. Germany has the cooperation of most of Europe, but none of the costs of occupation. The US won't pass Lend Lease to help the USSR alone. And if things get bad on the front, Germany can probably con Europe to send volunteers to the SS at minimum, or perhaps get them to declare war on the USSR to destroy Bolshevism. In such a scenario, it's possible a very favorable peace can be negotiated by Germany giving them most of what they want.