Things are definitely speeding up.
My timeline with a renewal at Peterborough is now:
Application made online: 28 April 2020
Old passport received at Peterborough: 29 April
Application being processed: 12 May
Application approved: 2.52 pm, 3 June (15 working days since processing initiated)
I was entirely reconciled to waiting until July or even longer so this was a nice surprise today. It isn’t over until it’s over, but I imagine I will have the passport within 5-7 days now.