This is odd; there is code in VisualEditor explicitly to re-login users in this situation. Maybe there's something that doesn't work with read-locked-down wikis, perhaps?

Yes, problem is on private wiki. In general I have issues with Mediawiki on account of this. I wonder how well it's been covered in any automated tests...

Interesting. I have persisted with this problem for 3-4 months.

I have mainly avoided using VisualEditor because of it or having to work around it by logging in using another window, click on "Switch to source editing" and then press save (with a number of Esc presses sometimes required to exit the Visual Editor save window)

Yesterday I installed Scribunto and Code editor and this problem no longer appears to occur. Is there a dependency here for e.g. a private wiki (or perhaps more generally)?

The only other change made recently was increasing timeouts and file sizes associated with the Special:Import and importDump.php script as detailed on the wiki pages.

And also, FWIW, the TemplateData extension