In general, thunderstorms hit in the afternoon. That means you have until 2 pm or so to go hiking etc. As others have stated, it’s variable this far out in time.
My one caution from my mountaineering days - Beware of thunderstorms sneaking up from the other side of the mountain, especially if you are on the leeward side. You won’t see the storm until you get to the top (the mountain is blocking the view). At that point, you are very exposed. Lightning is a big issue.
Thunderstorms also like to push up the valley. If you are in Grindelwald you most likely will seen the clouds start to collect for an hour or two before they do anything.
One suggestion: take the tram to the top in the morning and hike down.
Bring short sleeve shirts with add-ons for the mountain. One pair of pants that roll into capris are nice too. That will work for both places. A light cardigan works in the mountains and also late in the evening in Italy (and air conditioned places). The cardigan should be thin enough to roll up into your purse.
Mountain Add-ons are: a waterPROOF jacket, maybe a light baselayer top. Depending on weather, a puff jacket or vest isn’t bad either, especially if it rains a lot and gets bone chilling damp. A pair of light silk long John pants or a pair of leggings add warm under your regular pants. You can store them in a quart sized ziplock in your day pack. A pair of wool socks save your feet if they get wet.
In short, bring warm weather clothes and layer up for the mountains. There’s no need to bring two sets of clothes for two locations.