If I have understood correctly you are flying with two cats from the US to Paris, and then into the UK via Eurotunnel. The cats will then stay in the UK. You will therefore be required to meet both French rules and UK rules.
***Disclaimer: I am basing this on my non-professional understanding UK/general EU rules. The consequences of mistakes are severe (prolonged quarantine of your cats) so take professional advice from your vet, a government helpline, or a specialist pet travel agency. I would be surprised if BA would advise you on exactly what paperwork is required -- there's a lot of legislation here and it's a big risk for them to take.***
Rules for entry into the UK are on the link in your first post. The UK classifies the USA as a 'Listed Country' for purposes of pet travel. For entry into the UK (or, I believe, France)
directly from the USA your cats will need:
- a microchip. This will be read at the border crossing into the UK (and into France) to check that the microchip number (and therefore the cat!) corresponds to all paperwork.
- A rabies vaccination (with a specific inactivated or recombinant vaccine; your vet will be able to advise), carried out after the cats were microchipped. The last vaccine of the initial course must be administered at least 21 days before travel (the UK government says before 21 days before arrival, the French Embassy says 21 days before departure).
- A Third Country Official Veterinary Certificate. The PDF template for this is available via your link. This certificate must be issued by a vet in the USA (possibly a specific US Dept of Agriculture vet but ask your usual vet for advice) fewer than 10 days before your arrival into France. I
believe you can then use the same certificate to enter the UK from France up to four months after the certificate issue date.
This is complicated by your indirect routing, so I would contact the UK's Pet Travel Scheme Helpline to confirm my above advice:
Pet Travel Scheme helpline
[email protected]
Telephone: 0370 241 1710 [use Skype!]
Monday to Friday, 8am to 6pm (closed on bank holidays)
If you want to travel in December you will need to start this process ASAP. Some rabies vaccines require multiple injections weeks apart, and travel can't commence until 21 days after the last injection of the course.