If you’re not receiving the email, it means your hosting server has a problem sending emails, or the email is getting lost in transit. WordPress has no control over this (though you can install a plugin to use an external transactional email or SMTP provider with better deliverability.)
For a “backdoor”, you have two options:
1) You can make the change directly in the database, and no confirmation will be needed.
2) For an even quick “backdoor”, open the URL /wp-admin/options.php in your browser (note that you must be logged in with an admin account to view this, so this is by no means a security risk).
There you’ll see a whole lot of settings that are not normally exposed in the admin UI directly for reasons that should be obvious to you: you can easily brick your entire site here!
Look for the admin_email field which should contain your OLD admin email address. Change it to your NEW admin email address, scroll all the way to the bottom and hit SAVE… and you should be good.
After this, you may want to troubleshoot and address your website’s email deliverability issues. Here are some plugins that can help: https://www.ads-software.com/plugins/search/smtp/
Good luck!
