I live in Switzerland and renewed my passport about 18 months ago. You can no longer renew your passport at British consulates in Europe. I had to send my paperwork, including my old passport to an office in Paris (I believe it is an outsourced private company) which checks the paperwork. The passport is actually issued by the passport office in the UK and sent back to you from there. My renewal (time from sending in the passport to reception of the new passport) took about 10 days.
You can renew your passport before it expires (I think up to 6 months in advance) and the time is added on to the new passport.