When you set up comments on a blog post, wouldn’t it be lovely if they could all be held for moderation before being visible to all the visitors to your blog or site? It’d certainly be a good way to stop all that nonsense spam from cluttering up the discussion around your post and also reflecting badly on you?

Well you can!

It’d also be lovely to have an email alert whenever a comment is left so you can get straight on the case and moderate the comments.

You can do that too, but the notification emails don’t seem to turn up …!

Notifying people who comment that their comment is in a moderation queue? Notify them when the post has been approved (or not)?

Look here: https://wordpress.org/support/article/settings-general-screen/#email-address

In addition, if the option, An administrator must always approve the comment, is set in Administration > Settings > Discussion, this e-mail address will receive notification that the comment is being held for moderation. Please note this is different than the address you supplied for the admin user account; the admin account e-mail address is sent an e-mail only when someone submits a comment to a post by admin. The address you enter here will never be displayed on the site. You can send messages to multiple admins by using an email address which forwards email to multiple recipients.