If the cable service to your neighbour is routed through your property, you would have had to grant ntl an easement (legal right) for that cable (I assume you are the freeholder - if a leaseholder check the freeholder hasn't given consent). It very much sounds that you haven't done this. You are entitled to give notice to your neighbour and ntl that the cable must be removed and that if they do not do this by a specific date, you will do it. Serve such notice by recorded delivery (use your billing address if you don't have any other) I'd give them 7 working days from receipt of the letter.
I know you may want to retain good relationships with your neighbour but they should understand this is not a problem you have caused. They too need to get upset with the ntl installer.
As others have said, if you weren't happy with your service, I'm surprised you didn't complain before. I'm with ntl on this one that if they are unaware then how can they be expected to do anything and if they are unaware they don't have the opportunity to fix it so why should they then compensate for something they are unaware of in the first place
In any case there is not service level agreement for residantial services so there is no right to compensation in your contract, all you have is a right to terminate the agreement.