This (and US taxes) is one of the most stressful things about living in the UK.
The 5 year renewal comes around so quickly, and with our kids also being UK citizens, doubly so.
The hardest thing is getting an appointment during school vacation. The embassy releases dates about three weeks in advance and they are gone in an instant. A few years ago, we ditched our US vacation because we could not get US passports and met up with family in Niagara Falls, ON.
We did a passport replacement at the start of the Easter holidays for our DD. We had an 8am appointment, and got the photos at a place in Oxford Street at 7.45 am. We could have gone to a Snappy Snaps in our town but the appointment slipped up on me - but the Oxford Street place was a lot cheaper.
At the embassy, you queue outside so be ready for the weather. They check your appointment letter. You go through airport security where they check passports. If you have any electronics, they will take them for safekeeping. There is no need to travel without your phone or store it in a commercial place.
When you enter the building, you get a queue number. You sit in the waiting room until you are called. The first call is to hand over your documents. Then you go to the cashier. Then to reception to arrange the courier. After that, you have to wait a while for your application to be processed. At that stage, your family is interviewed by the consular officer, including the child, and then you are free to go.
Despite very stern instructions and forms, the whole experience is very relaxed and friendly. The passport takes about three weeks to arrive. I had my DD's passport delivered to my work and warned the receptionist to expect it. The tracking system pinpoints the delivery to a specific date, 8am - 6pm.
We had an 8am appointment and I think we were out by just after 9am.