Hopefully it will, but check your first bill that relates to the new contract carefully.
I have just renewed my contract. The plethora of Service Notices (most closed) and Questions (some open and some closed) on my account gave contradictory information regarding the terms of the renewal contract. The email I received, which is reproduced as a question, did, however, give the correct details but, for some reason did not specify the start date which was supposed to be my next billing date.
That bill has come and is wrong. The discounts appear to be those relating to my previous contract, and the My Product details on my dashboard have not changed to reflect the new contract. My conclusion is that the recontract has not actually kicked in as it should have.
Thankfully, I have a complaints team member monitoring my account due to previous ongoing billing issues and so I am waiting for him to sort things out and get my recontract up and running on the terms agreed and as set out in the email, and refund the overpayment on this first bill.
There appear to be too many problems associated with new and renewed contracts judging by the posts on these forums. The dreaded Plusnet billing system is possibly to blame but I also suspect some lax finger work by COT in my case judging by the content of one closed Service Notice on the day of renewal.