I had this problem when I was divorced, and relocated to where I live now, I first went to the post office to get a form, but was told there was no need to change it until the passport was due for renewal. I also contacted UKPA who basically gave me the same answer. I asked what if something unfortunate happened when abroad, UKPA said make up a slip of paper with your new details on, and stick over the existing page until renewal. This is also stated in the back of the passport.
When is your passport due for renewal, if it is within 9 months you can renew it now, and the time left on your present passport will be added to your new one. See the following web page of UKPA.