Just spoke to the IPS today on the phone for a similiar situation. I booked flights from Gatwick to Rome on the 24th May and then realised ten minutes after that my passport ran out a few days ago.
They said as it's May the queues (especially the London office) are getting larger and larger because of the summer holiday period. They could not guarantee that -although the average times at the moment were about three weeks- a new passport would reach me in time. Had to get the one day premium service and go to the next available appointment time next Thursday, 17th May.
Prices are as follows - £77 normal, £112 fast track (1 week turnaround), & £129.50 one day premium.
Remember though you need to have the application filled in whichever method you choose, so request one in the post anyway. If you need your passport within 4-6 weeks, do not fill it out online!