It did when upgraded to the sport package. I did have problems cancelling it online, had to phone BT customer services to cancel it, but that did that with no problems. Good luck
If you upgrade (add more content) it happens midnight the next working day, if you downgrade (remove content) it happens on the renewal of the pass/billing cycle.
When I upgraded from Entertainment Max HD to the Now TV Big Entertainment package with HD add on last March. The switch happened with in 5 or 10mins of me coming of the phone with BT. I just had to register my email with Now TV to get the on demand tuff.
I cancelled BT Sport at the same time but that toke at lest 10 days before they switched it off as I had to give 14 days notice for cancelling BT Sport so the guy said but I was not charged any extra from the point I cancelled BT Sport.