I mean all of your site cache, every cache you have active – plugins, CDN everything.
The search redirects to the correct URL, it is okay – the assets on those results pages are loaded from the wrong theme directory. For example when I search “Bambo” the script and the CSS files are okay. For example: https://babygreen.ch/wp-content/themes/organic_natural/js/hoverIntent.js?ver=20130729 -> This is correct
When searching “Pingo” some assets are loaded from the wrong theme directory. For example:
https://babygreen.ch/wp-content/themes/twentytwentyone/js/hoverIntent.js?ver=20130729 -> this is incorrect
These URI issues are completely unrelated to the plugin (it cannot change them), the issue is very likely the cache trying to serve a page, where the other theme was still activated. Once it is cleared and everything set correctly, it will be resolved.