MACs aren't used any longer. BT's terms and conditions (and other providers' as far as I'm aware) require 30 days notice for cancellation of broadband packages. That can be 14 days if you're switching to another provider, but I'm not sure whether that could work with a simultaneous home move.
We are moving to a new house and want our mac code to leave. We leave on the 25th August. Gave notice on the 15th August. BT say I need 3 weeks notice and the new residents can't connect their service until 9th September due to the notice period. They are on sky broadband so a BT line. What happens on the 25th will the new residents have our phone line for 2 weeks? That can't be right! BT say nothing to do with them! What a load of rubbish.
Sounds exactly right.
BT generally needs at least two weeks' notice, you gave them 10 days. BT's notice requirement is the exact same as Sky's, Plusnet's, TalkTalk.
You may find the only way of stopping the new people having access to your line will be to fully stop the service immediately and start a new contract at your new address. This will probably be more expensive than a standard move.