Auto copy value from specific user meta field to another field

You can simply hook function to sync phone number field as following;

<?php

// HOOK ON REGISTERING NEW USER/CUSTOMER
add_action('user_register', 'mm_sync_phone_number' , 20, 1);    

// HOOK ON PERSONAL OPTIONS UPDATE
add_action('personal_options_update', 'mm_sync_phone_number' , 20, 1);

// HOOK ON USER PROFILE UPDATE    
add_action('edit_user_profile_update','mm_sync_phone_number' , 20, 1); 

function mm_sync_phone_number( $user_id ) { 

    // GET PHONE NUMBER FROM OTHER FIELD
    $phone_number = get_user_meta( $customer_id, 'meta_key_of_other_phone_field', true );

    // UPDATE PHONE NUMBER IN BILLING PHONE NUMBER FIELD
    update_user_meta( $user_id, 'billing_phone',  $phone_number ); 

}

?>

This way every time, whenever admin is going to add or update user/customer from the dashboard, the value of your phone number field will be updated in the other WooCommerce billing phone number field.