My sympathies with your BT experience. From my own experience I know their customer service has fallen through the floor so far in recent years that it must be nearing Australia by now. I'm unsure why I still with them. Quarterly bills and no DD I suspect. And apathy I suppose. But they are terrible.
So your sister had a line and it was erroneously replaced by BT. I don't see why she had to open an account with BT in order to get that far. Presumably the other company was providing the line before.
You mean she was persuaded by BT to open an account with them ? I'm not sure I'd have gone down that path, it means she will be dropping her contract with the original firm. Not bad for BT eh ? They screw up and get a new customer.
Aye, e-mails provided by an Internet supplier will remain open for a short while for the ex-customer to move any e-mails they doesn't wish to lose, then the account would be closed. After all if you aren't paying to get broadband from them why would they continue to supply the e-mail service ?
Third party provided e-mails aren't going to care which ISP you are with.
So your sister was happy to pay Timico £130 to get away from them. I guess they must have upset her badly.
Meanwhile she can go on-line at a library or Internet Cafe.
To be honest it looks to me for all your attempts to straighten this out, all you are doing is swopping Internet Providers, and rewarding failure; at cost to yourselves. In your shoes I think I'd try to nip all this in the bud as soon as possible. Maybe get the ombudsman onto it ?