Have you tried direct connection to the tester socket SINCE your external cabling has been renewed ? Also, a different cable between modem and tester socket ?
Another thing you could try, to rule out faulty extension wiring is to try-out your dial-up modem to see if you can get a connection and if that works, then it would seem to point to a modem fault. It would be nice if you could borrow a broadband router/modem to test your connectivity.
As regards Tesco.net, suggest you have a look here, to see what problems others may be having :-
http://www.the-scream.co.uk/forums/f38.html