Further to my post above, my passport has now arrived.
Application made: 28 April
Old passport received at Peterborough: 29 April
Application being processed: 12 May
Application approved: 3 June
Passport printed and sent: 4 June
Passport received: 4.50 pm, 8 June (note that this is a Monday – a weekend slowed the delivery process)
Note also that although the PO and TNT had my email address and mobile number, I never received any tracking information in either an email or a text, despite the PO Advice Line insisting that I would do so. You may well need to contact TNT direct on email@example.com to get the crucial consignment number, without which you’ll get nowhere. Tell them your postcode and Passport Application Number.