Hello,
The Wishlist is working fine on your Site for the Logged In view, it is not showing items added to Wishlist in without Logged In preview of the Site.
Could you please once try deactivating all the third party Plugins on your Site, remaining the Theme required ones activated. If the issue gets resolved, then reactivate them one by one to see which Plugin is causing the issue. If this doesn’t work for you then once also, try to activate the Parent Theme on your Site if you are using Child Theme on your Site.
If somehow issue still persists, then please take complete Backup of your Site first, then provide us the permission to completely debug your Site which includes deactivating all the third party Plugins and removal of on-Site Customization changes too, so that we can thoroughly check this issue on your Site and assist you accordingly.
Best Regards