Hi rarecat -
My favorite trail in the Bernese Oberland that is both spectacular and easy is the walk from Männlichen to Kleine Scheidegg, which I usually access from Wengen.
In June 2015, I visited with my grandson, and due to schedule constraints, our trip was in early June. We saved the B..O, for the last days, and were up there on June 14 and June 15. The trail was open, with plenty of snow residue here and there. Unfortunately, we had clouds and fog for both the days that we were there, and finally ended up visiting Brienz.
Over the years, we have visited the Jungfrau region in early to mid June, and found that the high Alps were often inaccessible, or the cableways were not all running. By the middle of June,starting at about he 15th, tourist facilities and the transport options were opening up, and by late June, most things were open and the tourist were still not thick.
At lower elevations, such as around Lake Lucern, Spring was fully at work.
June is wonderful for the wildflowers in the alpine meadows.
We have switched to September as our preferred month. More stable weather, drier than June, and the tourist have thinned out, while the facilities are generally all still available.
Here is how I judge the arrival of Spring in the Jungfrau region. I have gotten in the habit of checking from late May through June day by day for the cableway service from Wengen ( Wengiboden) to Männlichen LWM (Lauterbrunnen- Wengen - Männlichen) in the timetable:
I use that as my personal measure of how open the higher trails are....that hike is at about 2300 meters.
I also check hotels for the dates when they open for the Summer season.
The pattern seems to be that the cableway up to Männlichen will be open on weekends in mid to late May, depending on snow conditions, and somewhere around June 15th it is usually open full time. Could be sooner if the snow has melted and the rail opened; could be later if there is a heavy snow pack.
This link does show trail open/closed status, but in mid-June, status my vary from day to day, and may not always be reflected in this chart:
www.jungfrau.ch/en/som mer/tourism/destinatio ns/kleine-scheidegg/hiking-trails/
As Annika pointed out, altitude is critical in the change of seasons.