How do I change my primary apple ID alias?

I have changed my Apple ID email address,

Previous Apple ID: old.email@gmail.com -> New Apple ID: new.email@gmail.com

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I have created an apple email alias, new.email@icloud.com

You need to change your Previous Primary Apple Alias: old.email@icloud.com -> to a new.email@icloud.com

Then see this Apple article: What to do after you change your Apple ID or password

What to do after you change your Apple ID or password - Apple Support

If you still need help, get more help and assistant from Apple directly, see Contact Apple for support and service -> Contact Apple for support and service – Apple Support (UK) 

I have created an apple email alias, new.email@icloud.com, but I can't make it my primary or delete the existing primary, old.email@icloud.com.

"Note: You can’t delete or turn off a primary iCloud Mail address. You can’t use an alias to create a separate Apple ID, and an email alias can’t be converted to a primary iCloud Mail address." - Add and manage email aliases for iCloud Mail on iCloud.com - Apple Support

I am not clear as to why you made an alias the same as your primary email, but oh well. See this eponymous article for how to Add and manage email aliases for iCloud Mail on iCloud.com - Apple Support