I shipped my cats separately from myself. Because I'd be shipping them into quarantine on the UK side, the quarantine (4PawsUK-- they're GREAT!) took care of the DEFRA side paperwork and sent me their boarding paperwork and crate stickers for the final Chicago --> Manchester leg.
I did try to contact DEFRA myself to get the paperwork in order, but DEFRA wouldn't respond to me-- I'm guessing they handle inquiries only from folks in the business... so you might contact PetRelocation or a specialist UK side to see if they will quote you for handling just the paperwork piece. I'm not sure if 4Paws could/would, but it never hurts to ask.
Then I took the list of approved route providers for the destination, and I phoned them for quotes and policies, because each one has their own policies as well.
For example, one carrier absolutely required that in order for me to ship my cats through them, the cats needed to be delivered to their carrier office in Chicago (I couldn't start their trip from elsewhere in the States through them and it had to be -me- not a courier in their office) no later than noon on the date of their flight... which would have been an impossibility for me.
I ended up going with Continental PetSafe -- of all the carriers I talked to, they were not only the least expensive ($1340 for both cats), they were also the most knowledgeable about how the process works. Their requirements for me included dropping them off 2 hours before their first flight and having their DEFRA and USDA paperwork in order.
USDA Paperwork (for the cats as they were flying through the US into Chicago before the final hop) included a Health Certification from my vet that needed to be sent to the State Vet for signature. This requirement might be different for you since you're in a different state, so give your vet a call to double check. My State Vet office also said that they needed to sign a EU Health Certification as well. 4PawsUK disagreed, but the State Vet was going to charge me $24 for signatures regardless of how many papers they were endorsing, so I went ahead and got both the US and the EU Health Certifications signed.
Customs UK side was handled by 4PawsUK who picked them up to deliver them into quarantine where the cats are sitting happily today.
Might be worth a think to go ahead and send the cats before you leave or as you leave and have them sit in quarantine for the remainder of their time on the PetScheme, which you can do. Instead of 6 months, they're just waiting the remainder of their wait time. 4PawsUK quoted us at ₤11/day for a single unit housing both cats and a ₤50 Administrative fee for all the DEFRA related paperwork. Another benefit of sending the cats ahead of you is that if there's something wrong or missing in their paperwork, you might have a little bit more agility getting it handled States side than UK side. Plus you wouldn't have to put this off on friends to handle while you're in the UK. I think it would be heartbreaking if something happened and the cats couldn't go and your family's separated =(
edited: my DH reminded me of a few items...