I agree it’s really frustrating.
The stats from this site show it’s taking them longer to process passports compared to last month, probably related to the lifting of COVID restrictions and demand to travel. So now it’s 23 days on average for child renewal https://www.passportwaitingtime.co.uk/child-passport-renewal-timeline/
When I was waiting for my passport, my backup plan was to re-apply (and pay again) for the 1-week fast track. This would have to start no later than 2 weeks before I needed to travel. You can only do the 1-week fast track, as opposed to the premium same-day service, because you no longer have the old passport, plus you need to allow some time for appointment availability (not always bookable nearby) plus possibility of delays in processing or delivery.
Good luck.