Hi,
I found a fix which I think is a bit of a hack. So for some reason Affinity is looking in "C:\Users\Andy\AppData\Roaming\Affinity\Designer\1.0\ProgramData", and it fails to write the license file.
When I run it in admin mode, it looks in "C:\ProgramData\Affinity\Designer\1.0" and succeeds in writing the license file. But this isn't good enough, because the next time I start it as a normal user it looks for the license file in the wrong place again and refuses to run. I realised that if I manually copy the "license.xml" file written in the ProgramData path over to the AppData path, it will find what it's looking for and work.
So there is still a nasty bug of some kind, but this hack seems to work for now.
Thanks for your help.