BOAC,
Oh the joys of deregulation and getting to who owns what.
When you put the broadband connection in - did you rent the line from BT & then get your internet connection separately? Or was it all in one deal from the ISP?
If you have a separate line rental agreement with BT (I assume it is BT as your first post says that BT did the tests) then the BT website gives you numbers to call if the line has a problem. By the way - if the line is one with a phone point, do you normally get a dialtone on it and if so then when your Internet connection fails in the rain, you can test for the dial tone as well as a check for line issues.
If it is an all in one agreement and you have to conduct business via the ISP support line, I would suggest writing to the management at the ISP and explaining (1) the issue you have and that (2) you cannot be paying 75p a minute to repeatedly log the calls as this is one continous issue and ask for a call ticket number & a phone number that you can call.
Let us know how you get on. At the risk of getting shot down - which ISP is it?