I've checked your Website on my mobile phone and switching the language using the "Language Switcher" in the burger menu works fine as shown in this video recording hidden link .
But, this is what I've done before checking, I disabled the "Maintenance Mode" and cleared your "CloudFlare CDN Cache". I think it's the cache issue you're facing and could you please check again on your mobile?
I've checked and it looks like the menu works fine when you're logged in. But when you log out, it doesn't work and I can replicate it on desktop too.
Could you please temporarily disable CDN and Cache on your Website? Or clear CDN cache? I've tried clearing the CDN cache on your Website but it doesn't seem to clear it. Could you clear it directly on Cloudflare and see if it works?
Let me know how it goes and I hope to hear from you soon.
I've configured the "Display Conditions" for your header template and it seems to work fine now. I can see the correct menu on mobile and desktop. Here is the link to the header template hidden link
Could you please check on your end to confirm? The attached screenshot is where I configured it.