Q: what do I do to my backup when I remove apple mail

Do you mean you are not suing Gmail apps vice just adding the Gmail account to mac Mail?

If that is the case just review your Time Machine Preferences>Options and make sure that the new app is not excluded or that it is included. How you check depends upon how TM was setup. The default is to included everything on your startup disk

Do you mean you need to update the email associated with your Apple ID?

Change your Apple ID - Apple Support

Note: if you have an iPad, iPhone or iPod, be sure to sign out of your Apple ID before completing this process. Sign back in when done.

Time Machine makes a copy of the files on your drive(s),  from all attached drives by default.

It does not change the choices you have made on your computer in any way.

Time Machine has nothing to do with this, do nothing.