Bit late, but I had a similar issue where my live mobile site did not have the language menu but within my editor and preview mode it was present. I was creating a site in English and French and it seemed that I did not publish the French version, only the English version.
A solution that worked for me:
- Make sure you’re in the desktop editor mode
- Go to Settings > Multilingual
- This should direct you to your Multilingual Dashboard
- From there, you can toggle what language site you want to be shown by clicking the ‘Show your site’ toggle button.
After doing this, my mobile website successfully displayed the multilingual menu icon and also integrated a language option into the mobile hamburger menu.
Hope that helps!