Hi Elizabeth,
I think it really depends on what you hope to see and do.
I have been living here for five years and as I look through old photos I can see what it was like on specific days of the year. I mention the flowers to help you visualize the season.
April:
In Thun (500m elev.) Spring flowers and cherry trees are blooming. There is often a dump of wet snow in early April that will melt within a day or two
Tulip festival is on in Morgues (you could visit this at the end of the Golden Pass journey, if you are interested). Spring flowers blooming in Montreux
https://www.morges-tourisme.ch/en/Z12055/tulip-festival
Lauterbrunnen valley:
Ski season runs until the end of April, depending on the snow
Many of the favourite summer hiking routes are also winter hiking trails as well
https://jungfrauregion.swiss/en/winter/see-and-do/activities/winter-hiking/
I have gone up the Schilthorn in early April, so beautiful!
Golden Pass route:
Sometimes you will still have snow most of the way, definitely the mountain peaks will be wonderfully white
I think this is one of the most beautiful times to go
If there is a very specific thing you want to do it will be good to check opening times, but if you just want to go up some mountains and see some beauty you will not be disappointed.
Here are a few of the opening/closing dates:
Train from Wengen - Kleine Scheidegg open until April 23, then closed for a short while
Männlichen Luftseilbahn running until April 10
Harderbahn in Interlaken opens April 7