Update user_login to change username

First:

The $wpdb object has the names of tables, with prefixes, pre-defined for you.

$wpdb->users == 'wp_users'
$wpdb->posts == 'wp_posts'
etc.

Second:

$wpdb-prepare() is essentially a WordPress aware printf, if you pass it more than one argument, you need to have some string/digit replacements %s %d

$sql = "UPDATE {$wpdb->users} SET user_login = %s WHERE ID = %d"
$sql = $wpdb->prepare($sql, $user_email, $user_ID);
$wpdb->query($sql);

Alternatively $wpdb does have an update method as well:

$wpdb->update($wpdb->users,
    array('user_login', $user_email), array('ID', $user_id),
    array('%s'), array('%d'));

http://codex.wordpress.org/Class_Reference/wpdb