the function wp_new_user_notification is one of the Pluggable Functions. In the documention you can read:
Pluggable functions were introduced in WordPress 1.5.1 These functions let you override certain core functions via plugins. The most up-to-date list of core functions that WordPress allows plugins to override is available at wp-includes/pluggable.php. WordPress loads the built-in functions only if they are undefined after all plugins have been loaded.
That means you can not override that function in your theme. Put it in an plugin and it should work.
Usefull links:
How to override pluggable function in theme? – wordpress.stackexchange.com
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