Thought I’d add what I learned from the process.
Once they scan the documents onto their system it can either be automatically approved in which case it says application approved, or it needs further examination in which case it says application processing. The automatically approved ones seem to take c.3 weeks, normally 2 weeks for the docs to be scanned into the system and another few working days for it to be printed and sent.
The application processing ones seem to take much longer, and previous posts seem to suggest 10 weeks from when they scan the documents into the system, longer if there is more questions or documents needed.
Only thing I can suggest is to ring up and be lucky to get someone nice enough on the phone who will give Peterborough a prod. I found if you be as polite and understanding with the passport helpline employees, that’s often mirrored in their assistance to you so they may go a bit further in helping.
Good luck and hope everyone makes their holidays!