I am receiving a message in the backend of my website that the WPML plugin cannot connect with wpml.org (but I've inserted the key; see screenshot attached).
There's a big chance that what you are seeing is a false-positive that remained in the server caching and the notification will be removed automatically when the cache will automatically clear on its own.
To check if that's a false positive, go to Plugins > Add new > Commercial > click the "check for updates" button and if the page refreshes and WPML is still showing as registered, then yes, that is just a false positive.