I've finally got my DS's first British passport... Our situation was complicated by the fact that we are temporarily in the US so he was born here, and we had to send his US passport to the UK for the British passport to be issued. Cue weeks of stress, the involvement of our old British MP and the British passport eventually arriving before our other documents were returned (including the US passport, which DS can't leave the country without), meaning we were effectively stranded here for even more time.
It was only when our MP became involved that we started to move forward. The online email service is manned by idiots - I asked various questions and just got lies and fob offs as answers. They repeatedly told me the application had been received later than it was, despite me having a FedEx signature for the day it arrived with them, they sent me an email saying I needed to wait another eight weeks THE SAME DAY as another department emailed me to say our docs were on their way to us and another email to say we'd have to wait when I had the British passport in my hand!
All in all, a total balls up and a total shambles compared with the US system (which should have been complicated given we are foreigners but wasn't!).