I moved mid-July and sent my new details off to the CAA. It took about 5 to 6 weeks to get a reply (I was beginning to get a bit concerned and very nearly gave them a call). You just get back a replacement page for your PPL, along with an updated RT licence.
It does takes a while for them to reply - strange, as it only took about 10 days to receive my initial licence. If you haven't heard in a few weeks, give them a ring. I don't trust the Post Office not to lose that small bit of card you have to post to the CAA.......