Try this -> Sign in to your Apple ID account page and edit / update your Apple ID details.
https://appleid.apple.com
- update your 'new' mail addres.
Keep in mind:
Your Apple ID is the personal account you use with all Apple products and services. It includes the email address and password you use to sign in, as well as the contact, payment, and security details that you use with Apple.
You can’t merge multiple Apple ID accounts.
About your old mail address, read this before you remove / delete it.
If you have deleted it, it is no longer valid, it simply no longer exists and it no longer works.
You will lose all apps, music and others that you have purchased with that old email address.
To understand about a deleted Apple ID, see the Apple articles below:
How to delete your Apple ID account
How to delete your Apple ID account – Apple Support (UK)
Understand and control the personal information that you store with Apple
Understand and control the personal information that you store with Apple - Apple Support