From what I can see, you are using the "dropdown" styling for the menu language switcher, so as you can see in my screenshot, you first need to hover your mouse over the language switcher to see all the language options.
I am unsure if this is caused by some settings from your current theme or because of settings added inside the WPML settings.
You can try to go to WPML > Languages > find the "menu language switcher" section > click the pencil edit icon > and from there, set the language switcher NOT to be a dropdown but to display all the flags on the same line.
Please let me know how that goes and if it resolves your issue.