Most ISPs have notice periods for cancellation - the OP should check with Zen on the procedure. From their terms and conditions it looks like it is only 14 days though. I would certainly advise checking with your old ISP after the changeover to make sure they have cancelled your account. In my case, when I changed from Virgin to TT, I requested a MAC to start their 30 day notice period (otherwise I would have been paying Virgin for a full month after changing to TT), but also had to phone after the switch to get my account cancelled (it was still active despite me being on TT LLU).
You should not need to contact BT at all though - you will probably get an "are you sure" begging letter from them before the changeover

. Just make sure you are not under contract with them still, obviously, otherwise you may face huge charges.