Hi Joannes. Welcome tomy Swiss Alps.
Wengen is my favorite town in the Bernese Oberland, and, another forum member has written extensively about it:
Unfortunately, April is at the very end of the ski season, so there are not a lot of customers in the area. It is "low season."
Many - but not all - hotels, restaurants and other facilities are closed.
Here is an example -
Picking April 13th and 23rd as random dates, the lift from Wengen(Wengiboden) to Männlichen ( Station Männlichen LWM) is not in service, and will not open until possibly some weekends in late May. An attached image shows the view from the cabin of the lift. Full service for hiking season begins in mid to late June, when the trails are re-opened at high altitudes.
The high trails are generally not passable. During ski season some are groomed for winter walking, such as the well known trail from Männlichen ( at 2230 meters elevation) to Kleine Scheidegg at 2061 meters. By April, they are generally not maintained.
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The cograil train ( Lauterbrunnen- Wengen - Kleine Scheidegg- GrindelwaldGrund does run all year, allowing access to the branch that goes from Kleine Scheidegg to Jungfraujoch.
On the floor of the Lauterbrunnen Valley, shown in the attached image from warm weather times, you can see the flat valley floor. It is only at about 800 meters elevation, and the trail/roads may be passable for walking.
Even in the lowlands, Spring is not far advanced in mid-April.
In 2014, in mid-April, there was a very unusual sunny and dry, warm spell over most of Switzerland. I was in the Emmental at a viewpoint above Sumiswald, at about 800 meters, looking toward Schloss Trachselwald and the Alps.
Images attached. The panorama was constructed with the aid of Photoshop, in order to combine several images and also visualize the Alps clearly as well as the foreground.
You will not that the trees are blossoming, but have not put out leaves yet. The grass has only become green in the warm and wet valley floor. The Alps have generous coating of snow.
In April, I'd look to the warmer areas with sunny slopes, such as the vineyards between Lausanne and Montreux.
I did a favorite hike in the warm and sunny Rhone Valley in April a few years ago, on the Südrampe Trail. I was early...I was there before Easter, when the trail as not officially open, and the winter debris had not been cleared. The handrails normally installed in some of the places where they are useful had not yet been set up.
If the trail is open when you are there, it is accessible from Spiez by the train that goes through Kandersteg. Get off at Hohtenn. It could be a "fall back" option for you.
Personally, i think I'd base in Luzernand try to walk around the lake, or else go to the Ticino or Montreux or Lausanne.