With apologies for my own ineptitude, this time with some html:
1) Check whether it could possibly be traffic from your network. Do friends have access? Children? Poor Wi-Fi security? Open proxy server?
2) Subject access request (s7 Data Protection Act 1998) to BT, for the record which Get it Right sent to them. You want the IP address, port number and UTC timestamp, which enabled BT to do the matching to identify your account. It might cost you £10.
3) If you are willing to identify yourself to Get it Right (more so than you have done by posting the letter online, I suppose), a SAR to GiR, asking for the information which they hold on you, which led them to send the notice to BT. If they quibble about whether they hold your personal data, point out the case of Breyer, dealing with IP addresses and personal data.
4) If you are sure that it was not you, once you've seen GiR's record, consider complaining to them about inaccurate processing of your personal data. Ask for rectification under the Data Protection Act. If they refuse, complain to the Information Commissioner's Office. They may not do anything but, the more complaints, the higher the likelihood.
5) You might even consider a small claims court action for the distress caused by their processing, if you can demonstrate inaccuracy. But I'd talk to a solicitor first at that point as, while the SCC generally shields from paying the other party's legal fees, that is at the discretion of the court.
6) Vote with your wallet, and switch to an ISP which has not voluntarily entered into this scheme. If you do this, tell BT that you've moved and why.
7) Ignore it, notwithstanding the frustration of its inaccuracy / inappropriateness.
(The last one I saw was a speculative invoice for a pornographic film. My advice in that case was to simply ignore it. Nothing further was heard. (YMMV etc.))