The less "touristed"area of the Engadine...nearest Austria, may also have a good deal of snow at altitude until or during early June. A hotel tha I used to use a lot - Piz Buin - in Guara, usually opened around June 1.
This is a good web site to check out trails and altitude. If you pick out a specific trail , you can get a lot of information, including detailed maps and an altitude profile.
map.wanderland.ch/?lang=de&route=all&bgLa yer=pk&resolution=50&X =675450&Y=200550&layer s=Wanderland
Altitude is everything when it comes to the weather and the seasons. The area around Lake Lucerne is more likely to be well into Spring even in May. I like that area a lot, have spent many weeks in the area or nrarby, and Annika has given you some links.
This link which includes some pictures, may give you some ideas.
The attached image gives some ideas of the topography around Lake Lucerne. It was taken from the top of Pilatus.. It does not have the high Alps of the Jungfrau Region....but, early June in the Jungfrau region only has low altitude hikes.
Here is some information about late Spring near the Jungfrau:
There wil be some pleasant hikes at lower altitudes, but not as stunning as the ones at 2000 meters, such as this one which rarely ( but sometimes) opens before June 15th:
You could check with the tourist offices in Interlaken and in Wengen as the date gets closer, to get a sense of what conditions are.
Many hotels don't open till early or mid-June, but rooms will be available on short notice. Not a lot of customers then. Skiing is awful, and hiking is not very plentiful.
I have attached some pictures taken in a warmer month....actually, mid-to- late September.
In both the Lake Lucerne region and the Jungfrau Region, the density of public transport assures you of frequent connections with various trailheads and hiking routes. So, you could pick a village around the lake, or anywhere in the Jungfrau region, and still get to various hikes. Of course, you are not permitted use a car in most of the Jungfrau region. Some of the regionaround Lake Lucerne is car free, too. Most of the Rigi is car-free.
Some time with the map and the timetable will give you ideas about how long various journeys take ( including lake boat journeys):
For Lake Lucerne (scroll down for the correct seasonal schedule and some details):
This map is good for travel connections. Use the menus. If you zoom in with "Trafic" turnd on, you will see icons for station stops. Mouse over them and you get a near term transport schedule.
You will find information on most sites in German, French, Italian, and English. No Flemish, unfortunately. ;-) However, Arno and Annika (moderators) are Dutch; that might come close. ;-)