I don't know if you have your place reserved on the flight for your pet but I'd do that the day that places open if you can. I believe it's 13 days ahead of time. I still have the email with the information if you haven't booked your place yet and wanted to look over it - it'll be sent to you once you book your place as well.
We went to the airport about 5 hours before the flight. We just had one of her towels in the bottom although apparently there is something that you can buy at the petstore for the purpose of transporting pets. The only other couple who had booked their pets on the flight arrived before us so we had to wait about 1 hour before we were even taken care of. I was quite nervous because the whole thing on the ATL side seemed pretty unprofessional. It took about an hour for them to check her in once we got started and then she was carried off and we just went to the airport for our flight. I had copies of *everything* because the airport people had to keep some stuff.
On the LGW side we landed and went through immigration and immediately called the pet people - she hadn't been unloaded yet or checked in. We didn't have the number for the pet people and had to get it from airport information...it's still programmed in our phone so I can look it up if you need it, it was worth having.
They recommended that we come to the LGW pet thing about 2 hours later (so about 9:30am). We got a cab there because you can't walk. I believe the cost was about £7-10 (can't remember) and I still have the number for the cab in our phone as well
Don't ask the cab to wait for you, it'll take forever!!
We sat in the main pet office for about another 45-60 minutes. We just had to give the receptionist all of our information and she confirmed that the animals had landed, been unloaded, and were being checked out by the vet. At about 10:30-11am we went about two building over to this warehouse/hangar looking place where we waited in a glass box for the vets or pet handlers to bring them out. That took about 15-20 minutes. Once the people delivered our kitty we could take her out of the box and feed and water her.
We called the cab and headed back to the airport to catch transportation links into London.
The ATL side seemed much less waiting than the LGW side. BA was great, as always, and their instructions were very straight-forward in the email and on the phone.
I'm not sure that's the great timeline you were after, but hopefully it helps. I didn't find the timeline tight at all. We were still in ATL about 3 hours ahead of time and checked in with about 2.5 hours left to go. I'm also guessing that you could get there a bit earlier, the couple in front of us must have been quite early considering where in the process they were.