This thread is resolved. Here is a description of the problem and solution.
When adding a product to the cart in the other languages, there is a message on the cart's page: "Session expired return to the shop."
1. Go to WPML -> Languages.
2. Scroll down to the 'Language filtering for AJAX operations.'
3. Check the checkbox 'Store a language cookie to support language filtering for AJAX' and save.
4. Go to the front-end and check if it is working correctly now.
This is the technical support forum for WPML - the multilingual WordPress plugin.
Everyone can read, but only WPML clients can post here. WPML team is replying on the forum 6 days per week, 22 hours per day.