The US had a panic attack when France collapsed, mobilizing every reservist & National Guardsman & setting in motion ramping the Army to 1,600,000 men in eight to ten months, along with handing the Army & Navy a blank check for purchasing equipment. When Japan grabbed French colonies in Asia the US threw a conniption and led in the imposition of economic actions that came within a centimeter of being a DoW. So, if Italy tries the same stunt with Tunisia? I'll leave the readers here to judge for themselves.
They will do between nothing and nada; Japanese troops in Indochina are a danger for the US interest and a strategic problem...Italy grabbing some French colonies in Africa are almost at 'who care?' level with an hint of 'where the devil is this place?'; plus while the Wallies had basically accepted the fact that in the end Italy will have entered the war at German side and put military and economic pression on her, the Fall of France changed everything.
The last thing that Churchill and everyone in the UK will want after Dunkirk it's the Italian dow, so while them grabbing some colonies will not make anyone (German included) happy and it's very probable that Adolf will put some pressure himselfs to contain Benny, in the end Italy will have a relative free hand (expecially if London can use her to keep the balkan free of German and Soviets)