I have four browsers (Firefox, Safari, Chrome and Opera) and this feature does not work on ANY of them! I have exactly the same problem as the OP - FamilySearch signs in fine, but Wikitree ignores me every time and just keeps presenting the log in (which never works).
This means that I can never use this feature - it's a real sticky irritating little bug? Surely some Wikitree techie can fix this - it's rather a ridiculous state of affairs. (I have only tried this on my iMac, not on any mobile devices. I assumed it would be foolproof - not so!)