I think I called the USDA office (New York state) about 3 weeks before to the 10 day window. I didn't fax any paperwork up to them (wasn't a requirement for this office) prior to the appointment, as I was close enough that if there were any problems I could run back to the vet and get it corrected (luckily I did not have to do that)!
Why not fax (or have the vet fax) the paperwork to the USDA office first thing today and get the ball rolling so you can get the appointment set up? It will accomplish two things, get the appointment set up and if there are any errors on your paperwork you will have time to get it corrected. Can you call the office or do they only accept emails? If you can call it might be easier to get it all sorted then going back and forth with emails...
GAtoWales, it is easy for me to say (now that it is a year down the road) but really, try not and stress too much about all of this! The paperwork is really just filling in the blanks and doing all the correct steps in the correct order. I know it can be awful as you are dealing with your own vet, the USDA vet, then the airlines and it is a nightmare at times trying to get it all coordinated! As long as you have done the microchipping/rabies shot/tapeworm all in the required time and have the forms filled out and signed by all involved (remember
Blue Ink Only on the forms)! You and your dogs will be A-OK! If the worse comes to worse comes to the very worse, the dogs might have to stay in a kennel at Heathrow? (I don't know which airport you are using) for a couple days until whatever needs to be corrected on the forms is done. But that is like the very worse of the worse scenario! It won't happen!
To help ease your stress, you can also fax or scan & email your paperwork to the pet reception office at whichever airport you are flying into, it is often done and from what I have read they seem to be a friendly bunch at these pet reception offices and will check it out and let you know if it is ok. I would do that if I were you, will help calm your nerves for sure

You and your dogs will be fine, you seem to be on top of it all and before you know it, it will all be over with!