Hi @Deanc,
I applied for my passport on 19/5 and was going thru the same exact situation as yours. I must have called over 30 times over the past few weeks, and based on my experience nothing is done through them because when I finally thought that they noted down my travel date, the advisor from yesterday’s call said it’s not showing on my application even though many before him have claimed to have noted it down.
I think the only thing that helped my case was sending emails to all the upgrade teams (someone posted the emails before on here), and in the subject line, write Urgent, in breach of SLA, your child’s full name and ref no., over 10 weeks processing time and travel date. Don’t forget to attach the tickets as evidence. In my case, the upgrade teams couldn’t help with my application as it’s a first time adult passport but the one in Belfast advised that they will contact the expedited services team as it’s been over 10 weeks and that they will treat it as a priority. They sent me this email yesterday and lo and behold my application got approved today.
Just keep emailing the expedited and upgrade teams every 2 days with the same subject line. I believe they will approve it nearer to your travel date, as today marks 6 days before mine. Your other option is to wait 48 hours before your flight and go to your nearest passport office. I also emailed my MP who contacted them and sent complaints to all emails I could find. I personally think the subject line is very important as it’s the first thing they see while going thru so many emails.
Sorry for the long post but hope this helps. Good luck.