Relieved to say that our child renewal passport arrived today. Timeline as follows:
Tue 27 May - posted via special delivery
Wed 28 May - received and signed for in Glasgow
Started chasing on the helpline after about 2 weeks. Got nowhere.
After 3.5 weeks (Sat 21st June) I emailed CEO and our MP just asking for someone to tell me everything was "okay".
On Monday 23rd, MP responded to say she'd spoken to them and the case was marked as urgent, and to wait until Monday 30th before chasing again. We don't travel until 11th July, so that was okay.
Wed 25th - Glasgow office call to query a minor spelling mistake on application form (in middle name of child) - but they were happy to re-issue based on the name they had on the system from the previous passport (thumbs up for common sense). Said passport would be "issued" and to allow 3-5 days for delivery.
Thu 26th - passport arrived.
So, all in, 4 weeks and 1 day from date the documents were delivered.