I've read the weather in these area's can be hard to predict. Is late May, early June a good time to attempt to see these area's? I have my heart set on taking the cable cars to the top of the mountains in each regions.
Gosh, Steve, last time you posted those photos from Hallstadt you said it was April, not late May.
It is not just the weather, but the actual conditions on the ground that will affect your visit. But if riding lifts to the top (and back down) is your goal, rather than hiking, you should be OK.
You can see what it is like around Zermatt from the webcams:
I looks like it snowed there yesterday.
The Zermatt.ch website will have schedules for the Klein Matterhorn cablecar. It should be running, as they are sill skiing up high there.
In the Interlaken area, some of the high lifts (such as the Mannlichen lift above Wengen) will no open until the end of May---so June should be OK. You can check the schedule of the Schilthornbahn on its own website, www.schilthorn.ch
The Jungfraubahn railway will be running---it is always snowy at the top there.
We arrived in Zermatt last year on May 25--coming up from Italy. We were very fortunate to have perfect weather and had a view of the Matterhorn from our hotel balconey. While in Italy we met people who had recently been in Zermatt for several days and never saw the Matterhorn due to bad weather. Zermatt had very few tourists when we were there and it was a perfect visit in every way. But, OTOH, the weather cannot be predicted. As another poster said in the BO has more options on alps. We spent six days there and never made it to the Schiltorn due to fog but the Jungfrau was great. Some lifts do not open until June 1.
Weather won't stop the cable cars from running, but nobody can guarantee you'll have a clear view at any time of year. Late May and June are better than April, but not as good as the late summer and early fall. The best you can do is keep a flexible schedule.
And even though Rick has decreed Zermatt a "one mountain town"... that's far from true.