If you type www.tthfanfic.org then log in everything is fine, cookies are properly saved, etc.
If you type tthfanfic.org you still end op at the site, and if you are already logged in things are fine. But if you log in after getting to that URL the cookie generated is invalid and is properly rejected by your browser.
What I think happened is that the first time I went to the site with the new PC I typed tthfanfic.org then created a speed dial button pointing to that address, not www.tthfanfic.org - the same with the iPad and with the other browsers I tried. So every time I tried to log in it failed to stick, and because the address appeared to be correct it wasn't apparent that that was the cause.
I'm not sure why Chrome worked - I suspect that it either automatically added in the www, or didn't check the cookies properly, but I haven't checked which yet.
The reason for this is that whichever system generates the cookies truncates the URL from the front - if you use www.tthfanfic.org it's in the cookie as tthfanfic.org, which is a plausible URL, if you use tthfanfic.org it's stored as .org, which is an invalid URL, so the cookie is rejected.
There's probably a lesson here about laziness or something, but I think I'll just go with being pleased that things are finally working properly.