After the upgrade my blogs were not available any longer. It looked like a serious database problem. I felt in disaster mode for days (because I went on a holiday and I did not immediately have the time to solve the issue). It wasn’t really a big deal, but I wasn’t able to find this problem description (together with a solution on the web). The error message showed the blog URL, that my WordPress was looking up on a database table, and the letter case appeared strange to me. That was finally the key to the solution. See below!
I first tried the offered upgrade button. That lead to the “disaster”. Later I also tried the manual upgrade, i.e. the manual unpacking of the .tar.gz file – I’m not sure, that was necessary. Next time I shall try without that.
My PATH_CURRENT_SITE makes use of upper case letters – that seemed to disturb the seemless upgrade. Just for completing the upgrade I lower-cased it temporarily. After that completion I “uppercased” the previous letters again, and it was all fine then. Temporarily “lowercasing” that specific value withing wp0.wp_blogs didn’t seem necessary either.