Panic over. I was following it on a couple of forums for a few minutes and towards the end of the threads people were starting to say "I did x, then y and then z and now it works" - with x, y and z being different things. With the "Oops, we can't seem to find the page you requested" at apple.com that came up when it failed to connect, it started to look like it was somehow fixed at Apple's end.
Sound a bit sinister, that your phone reports back to Apple every time it tries to connect to your home wi-fi? Not really - apparently it tests that it can connect to the network by getting a webpage on Apple's domain that consists of a single word ("success"). If it doesn't find it assumes it can't connect to wi-fi and needs to ask for the network password. In the meantime it shows the "Oops..." page it now got via 3G (using iOS 6's new "wi-fi plus cellular' function where 3G takes over if a wi-fi network is flaky) because the "success" page was down. Fascinating!