The best time to see wildflowers in the alpine areas is early and mid-summer.
Rockoyster mentioned Wengen as a base, and I would agree with that - it is a lovely small alpine village with stunning views, and where traditional chalet-style buildings are the predominant building style.
A company that runs wildflower-viewing tours from a Wengen base runs their tours between 17 June and 1 July, so I expect they know from experience that the second half of June is the best time to see wildflowers in that area. Given the amount of snow that has been around this winter, I would expect your best chances this year would be right at the end of that period, rather than right at the beginning, but this is just guesswork on my part.
This report from a participant on one of these tours might interest you, and assist you to find the best spots for viewing the flowers:
www.rps.org/adminuploa ds/external/images/nat email@example.com/Alpine%20F lowers%20of%20the%20Sw iss%20Alps.pdf
There is also the Allmendhubel Flower Trail, accessible by funicular from Mürren (on the opposite side of the Lauterbrunnen Valley from Wengen):
jungfrauregion.ch/en/s ummer/product/page/all mendhubel-1907-masl-768/
The summer season for the funicular to Allmendhubel runs from 23 June until 14 October 2018.
Another good excursion would be to the Schynige Platte Botanical Alpine Garden, which is reached by a stunningly scenic train ride from Wilderswil (on the rail line between Interlaken and Lauterbrunnen). The Alpine Garden reopens after the winter closure on 26 May 2018, and remains open until 28 October inclusive. The train from Wilderswil to Schynige Platte is also expected to open on 26 May, but weather could delay it a bit.
Depending on how long your stay is, I would not necessarily change bases, as you will find plenty to do in the Jungfrau Region and nearby if you will be there from mid-late June onwards.
If you really did want another base, I would suggest Luzern. Apparently the Rigi is excellent for wildflowers, but I haven't been there at the height of the wildflower season myself.
PS My forum nickname, Alpenrose, is an alpine flower (but a rhododendron rather than a rose), and you can see a picture of it at the link below. Many hotels also have the name Alpenrose or Alpenrösli (which is the Swiss German diminutive form of the word).
https://www.adelbodner .ch/assets/Fotos/Hinte rgrund/alpenrose.jpg