I just moved my two cats from USA to London last week by booking directly on IAG/BA cargo. It was about $1500 total for the flights, plus around $400 for the paperwork from my vet (not including rabies/vaccine boosters since they were due anyway, and both were already microchipped).
I found the process to be relatively straightforward and IAG/Heathrow Animal Reception Centre were both very good whenever I called with questions or to verify I was on the right track.
I would recommend doing whatever you can to find a vet with experience completing the paperwork, as my vet did say sometimes a small typo or error can cause a big problem, but I am glad that I didn't spend extra on a shipping company just to help me do the paperwork because it really wasn't that bad. I moved from NYC so didn't have trouble finding a vet, and once the paperwork was completed I called Heathrow with the few questions I had to verify exactly what they were looking for. The paperwork was also looked over and approved at the airport by the usda, so I felt pretty confident by the time that we were actually traveling that they wouldn't have a problem getting in. The whole thing was a little nerve wracking, but went very smoothly in the end and we've all arrived without issue!