Spring does not come early to the Bernese Oberland. Many of the higher elevation hikes are often closed by snow and/or mud until mid-June. During heavy or late snow years some of these trails don’t open until July 1. The earliest the higher elevation hikes open are when the transports to reach the trailheads are scheduled to begin operating. However, sometimes the trails don’t open for days or weeks later, it is entirely weather- and snowpack-dependent.
However, there are a number of lower elevation hikes that are usually open during April, May, and early June. These are all easy walks; the more challenging hikes are those higher up.
A beautiful valley walk begins in Lauterbrunnen and ends in Stechelberg (or earlier, depending on your inclination). It is about 4 miles long on a wide flat path. You can go as far as Trümmelbach Falls (about half way and highly recommended for a visit) then bus the rest of the way, go as far as the Stechelberg Schilthornbahn station, about 3.5 miles total, or go all the way to Stechelberg where there is a lovely inn that serves delicious lunches. You can take the bus back from any of these stops to Lauterbrunnen.
The only walk at elevation that's pretty sure to be open is the Grütschalp to Mürren route. Take the cable train from Lauterbrunnen to Grütschalp then enjoy the 3.5 mile, mostly flat stroll to Mürren. Stunning views of the Eiger massif from here. You can also walk from Mürren down the paved easy trail to Gimmelwald then catch the gondola down to Stechelberg.
The Allmenhubel lift (from Mürren up to Allmendhubel) generally opens the end of the first week of June, walks in that area may be open that early or maybe not, entirely depending on snow levels. schilthorn.ch/en/Angeb ot/Timetable__Tariffs/ Timetable
The gondola from Wengen to Männlichen (where one begins the easy walk down to Kleine Scheidegg) is usually scheduled to begin operating in mid-May, but often the trail is obscured by snow or mud until mid June. If it is closed it is still well worth taking the gondola trip and walking around the top.
There is also a lovely easy walk along the river (starting in Lauterbrunnen) called the "Natur und Bahn" trail. You pick it up right behind (east) of the train station and follow it as far as you want. It runs to Wilderswil via Zweilütschinen, each of those segments is around 3 miles. You can catch the train back to Lauterbrunnen at either spot.
There are a number of loop trails around Wengen that are usually open before mid-June as well. The tourist information center in town can direct you to these.