Trying your modem on your neighbours computer shouldn't work, anyway, so that is a red herring, in my opinion.
I'm not sure why technical support are not checking if they can 'see' the modem after you have power cycled it? - have you tried logging in to the modem status pages ?
http://192.168.100.1/
username & password are both root
(hopefully that works with ex telewest accounts)
The webmail is more worrying, as that should be available from any connection, has anything else changed, recently? Not sure about blueyonder, but sometimes on ex NTL changing your password doesn't always automatically 'propogate' to other parts of the service, e.g. webspace access, which will often still want the 'original' password.