The page at https://confluence.atlassian.com/fisheye/fisheye-upgrade-guide-298976988.html#FishEyeupgradeguide-2.BackupyourFishEyedata has notes but they aren't very helpful.
Step 1 talks about uninstalling without explicitly stating that you have to. I found when I ran the installer it told me I had to uninstall the previous version first. I did that but when I ran the installer again and followed it through to Step 8 it left me with a system which didn't work. The whole thing is on a VHD so I can revert fairly easily but that's not a useful result, obviously.
At the moment the sequence seems to be backup, stop the service, uninstall the previous version, install the new version but point it at your existing data folder. It's supposed to run automatically but doesn't, I had to start the service manually and even then it didn't work, the website is unavailable. My data folder is surviving the process, which I wouldn't really call an upgrade, more of a reinstallation. Have I missed the point somewhere?
In case someone else finds this looking for an answer - you need to uninstall the old version before you install the new one so it isn't really an upgrade, more of a replacement.
I also needed to add the database connector to the new installation, something which became evident once I spoke to support and we checked the log files.