Not really probable, since by 1939 Mussolini was firmly entrenched on the Germany's side.
Anyway, let's suppose that Mussolini, enraged by the Molotov-Ribbentrop pact, harshly condemned as a "betrayl of the fascist ideals", makes a U turn and side Italy along the allies.
This opens some interesting scenarios. Germany could be in difficult situation, since she would have another front to defend and the italians could try to advance in September-October 1939 to occupy some position. In 1940, Hitler would probably proceed to the attack on France, while staying on the defensive on the italian border.
Now we have two possibilities: France falls as OTL, or the german need to man the italian border gives to the french the possibility to resist. I find more probable the first one, but the second is far to be unrealistic.
If France falls, the fight would move to the Alps. The mountains favours the italians, since they had fortified the border, had a huge experience in this kind of warfare and the impervious landscape nullify two of the main german advantages: tanks and airplanes.
At this point the war could become a rehearsal of WW1, only with the italians in a really more favourable position. The british would send all the troops available and few french division could try to seek shelter in Italy, rather than capitulate. Even the french mediterrean fleet could decide to defect (OTL, they scuttled the ships, IIRC). On a side note, probably there would be no Battle for Britain, so the british would have an easier time to reorganize. Greece, also, would be untouched by war.
The whole scenario now would play on the allies resistance on the Alps. If they manage to halt the german advance until the US enter war (there's no reason to think the pacific theatre different as OTL) and maybe Russia too, Hitler is defeated in 1944. If the germans breaks in, they could have an hard time anyway: as Kesserling showed, a determined defender can make life really difficult for his enemy in Italy. On the political side, the king could be tempted to betray the country as he did OTL, but I consider this possibility less than probable, since the royal court could alway relocate in Sicily or the colonies, under the allied fleet protection (so no back on the wall sindrome). Mussolini, on the other side, has no other alternative than fighting till victory, in this scenario. A defeat would mean death for him.