Data transfer to new phone defaulting to wrong Apple ID email address

I have backed up my old iPhone to iCloud and I am trying to transfer from my old device to my new iPhone. As I go through the process with the new phone holding it beside the old phone, the new device eventually asks me for my iCloud account password but the iCloud account e-mail address shown is incorrect and is one that I have no access to. I want to make sure that all my apps and data transfer across to my new device seamlessly and that I lose nothing in the process.

How do I make sure that the new device is picking the data from the proper iCloud account that I have backed the old device up to? And how do I remove the wrong e-mail address being suggested so that I am prompted with the correct iCloud email address and asked for the password associated with that?

Again, I need to ensure that I don't lose any data in the transfer.

Thanks for any help!