OK, so I went ahead - and it has not ended yet:
Issue 1: The Fairphone updater only showed the Cherry 1.6 update.
Issue 2: I had to tell the updater several times to start the update (maybe something was downloading?)
Finally, it went ahead.
It looks that it then updated the updater itsefl ;-), because now the Kola Nut 1.5 update was avaliable.
I went ahead and installed the 3 parts according to the instructions at New software update Kola Nut 1.8.5 is ready
After going through all 3 update steps (the last one asking for my Google credentials).
Issue 3: Clicking the Play Store app or GMail: “App isn’t installed”. Google Maps did start. A reboot did not help.
Well, I had still to install the Storage Upgrader which wipes all my data anyway, so I first went ahead with that.
Went into the Fairphone updater, Advanced Mode, and started the installation of the “Storage upgrader Kola Nut 1.8.5” which was under "Older versions"
Then I was (again) presented with a subtitle “an older version of your current operating system”, so I aborted.
The weird thing is: If I tap on the ‘Latest version’ (1.8.5) the next screen also says ‘older version’.
So what is going on here? Do the FP interfaces say ‘older version’ (instead of ‘current version’) if it is not a ‘newer version’? That would be sloppy.
Can I go on with installing the ‘older version’ storage upgrader?
[Pending an answer on this last question, I will now try to fix the Google issue.]