Try to find %Appdata\Roaming\Mozilla\profiles.ini file and check references to your profile. If you've deleted the profile, you can delete the profiles.ini eventually.

Thanks, but I said I'd done that. It is a fix already in the forum files. After doing that (searching AppData) I actually searched the entire C drive for profiles.ini and ... not there. I think it must be some registry entry? Does someone who writes Mozilla stuff not scour these issues?

I've always used the default location for my profiles, so I don't know whether there is a registry key associated with non-default locations. If you are familiar with RegEdit.exe you could check keys such as:

HKEY_LOCAL_MACHINE\SOFTWARE\Mozilla HKEY_LOCAL_MACHINE\SOFTWARE\mozilla.org

HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Mozilla HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\mozilla.org

I think this one is less used:

HKEY_CURRENT_USER\Software\Mozilla

Found entries there (as above, thanks jscher) and deleted entries entirely, and any others by searching mozilla/firefox/thuderbird but still I get the message that the profile is unavailable.

FIX for me: I had a large disc 'E' drive for storage that I renamed 'D' and Thunderbird now goes. I could not find or get Mozilla to forget that it had an installation on D drive so I had to provide it with one. Worth a look Mozilla?

Firefox and Thunderbird both place their Profiles (user data) on the drive where the operating system is loaded - typically C:/ - regardless of where the user installs the Firefox and Thunderbird programs.

If you have an operating system on D: or E:, that is where Firefox and Thunderbird data files will end up. You need to know what you're doing when you do that. Mozilla doesn't keep you from doing that, but they aren't concerned about "looking" at user done changes / modifications or non-default installations.