I know this is an old post but I'm sure a lot of people will be interested.
Short answer is no, Royal Mail do not have a public API. They do however have some kind of API they allow PayPal to use, since you can print postage through PayPal and their dispute process appears to check the delivery status of a package immediately after the tracking number has been submitted.
phpBB is way out of date and it's full of spam so it's clearly not being maintained. I called them yesterday to ask about the API as that post indicated one was under development a few years ago. I got through to the Royal Mail website helpdesk, who put me through to the department that dealt with maintaining the website. They had no idea what an API was or of the existence of this beta site. Incredibly, no one I spoke to had access to any website other than www.royalmail.com