It's never been easier (I hope)
One of the (many) reasons I made the trip back to the UK, when I did, was my passport was going to expire.
Today I headed over to https://www.gov.uk, searched for 'passport' and chose 'renew'.
I answered some simple questions, uploaded a photo Julie took of me with my iPhone, and paid the fee. Tomorrow I mail off my old passport. Within 11 weeks I should get a brand new one back.
This beats the experience a decade ago of getting multiple photos taken in the USA (to UK passport standards), having them signed (thanks Jeff), filling in forms and mailing the whole packet off with a cheque.