Hi,
I'll be flying on BA along with my dog to Heathrow in early July. This Saturday, my vet plans to issue the paperwork that we need.
What I found most helpful/least confusing was to just call the ARC at Heathrow, and go over the paperwork requirements with them. Another resource is JCS Livestock, the Customs agent used by BA. I emailed with a lady there today, and she confirmed the documents list that was laid out for me by the ARC clerk:
Defra Annex II form, signed by local vet & USDA vet
APHIS 7001 form, aka International Health Certificate, signed by local vet & USDA vet (so says the USDA vet I spoke to last week)
Microchip certificate
Rabies certificate - USDA vet also wants this sent to him
C5 Customs form - BA will send your contact details to JCS, and JCS will email the form/instructions to you - it's very straightforward, and gets emailed back to them prior to flying; this is what they use to clear your animal through Customs, prior to the ARC releasing the pet
A couple of caveats; I spoke to 3 different USDA people, and the last one happened to ask me if I was flying on the same plane as my dog; at the time, I didn't think I would be. He said that Annex II is only for "accompanied pets," whereas Annex I is for "unaccompanied pets". Annex I has a smaller window for tapeworm treatment. He offered a workaround option, something along the lines of filling out both forms, having both signed and abiding by the tighter tapeworm deadline, but in the end it worked out better for me to just be on the same plane - so I'm OK with Annex II. *He could have been totally wrong* throughout all of this research, I've gotten many conflicting answers, which is why I say it's easiest to just call the ARC and get the info straight from them.
Also about Annex II, be sure all of the dates are within 10 days of ARRIVAL in the UK, not 10 days of check-in at the airport. The tapeworm treatment sign-off is confusing; the USDA vet said technically it's supposed to be signed off on him AFTER the tapeworm treatment is given, but since most people do not have enough time to work with to get the form to the State's vet, work within their turnaround time and then have it back in-hand before they fly out, he told me to "listen between the lines" and have my local vet fill that portion out with the date he intended to do the tapeworm treatment, and the USDA vet would sign off as normal. By the time the tape treatment had been done & signed by my local vet, nobody on the UK side would be the wiser. A bit tricky and not really in the spirit of the form, but a real-life solution to a very tight timeframe.
APHIS 7001 also has a 10-day validity window if it's also being used as an Airline Health Certificate, which BA does require (same form can serve as meeting BA's requirement as well as filling the International Health Certificate requirement, but only if it's issued within 10 days or less of travel, as per USDA's website)
Both the ARC & JCS Livestock offered to review my paperwork for me before I head to the airport; I think the ARC sometimes pre-clear paperwork to make processing go a little quicker when the animal is actually there, so it's helpful to them too to have it in advance.
Hope some of this is helpful; I'm nervous for my dog's sake also, but I look it as just something we have to go through to stay together. Some of the rules may differ for dogs vs. cats, but since I'm days away from starting the paperwork you were asking about, I wanted to reply

Good luck!!