So, you've activated your SIM, given your PAC to giffgaff and are waiting for your number to transfer.
You've gone into your banking site/app and thought you'd cancel that DD as you'll no longer be needing it and you'll settle your final bill via customer service or the website with your credit/debit card when it coes in right?
Once the transfer completes it should indeed cause the billing system to generate a final bill, however bear in mind the billing process is pretty much completely automated and if the billing system is unable to collect payment via the Direct Debit mandate you agreed to it may well slap a default on your credit file.
This can be time consuming (it can take months) to get removed as you usually have to e-mail or write to a credit file team as they aren't usually contactable by phone and can take a while to get to your case.
Meanwhile you might have problems with obtaining credit (loans, overdrafts, cards or even mortgages) so it really is best to let the system collect your final bill via DD.