The agent fee was CNY1600, the UK Passport Office fee was GBP139 (for the 54 page passport)including their courier fees to and from UK. So it looks like the fees are similar to those of the US.

@Bothroyd. This answer is probably too late for your, but for other UK applicants it would be relevant. I waited until succesful completion of the process before posting.
You are correct, applications for UK passport replacement or renewal must be done at a British Consulate or the British Embassy, and it must be done 'in person', it cannot be done online. However, that 'person' can be a third party.
I have just done this. Basically I searched online for every visa agent in Shanghai, online. Several listed already appear to have a dead website or dead phone. Out of the 6-7 I contacted to ask if they could do a passport, only one replied (Shanghai Softlandings HR Co. Ltd.) and they said that they could do it.

I sent my passport and other documents to the agent, this included an authorisation letter for them to act for me. The agent submitted the application to the passport office (PO) in Shanghai, the PO then forwared every thing to the UK. The new passport was sent to the PO in Shanghai, and this was then collected by the agent who couriered the docs to me. Six weeks after initially posting my docs to Shanghai I recieved my new passport. The total cost was approx 3400 rmb.

I had my flight, for next week, cancelled this afternoon. We proactively called the airline (all in Chinese) and got put on a flight earlier on the same day. There was a long queue on the phone system, probably because everyone was calling about the same thing.
It looks like it was a case of cancelling half full flights, to fill other half empty flights. That may, or may not, have been the issue with the cancelled flights to Beijing, if it was, you need to get on the phone immediately and rebook for the same day if you can.

As you have plenty of time, it would be better to find a time to pop into the PSB/visa office in Tuodong Lu, and enquire there. That way you will have a difinitive answer. In the past I always found them to be extremely helpful.

Edit. Any office hours should have been stated in the contract...